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Essays About Math: Top 10 Examples and Writing Prompts 

Love it or hate it, an understanding of math is said to be crucial to success. So, if you are writing essays about math, read our top essay examples.  

Mathematics is the study of numbers, shapes, and space using reason and usually a special system of symbols and rules for organizing them . It can be used for a variety of purposes, from calculating a business’s profit to estimating the mass of a black hole. However, it can be considered “controversial” to an extent.

Most students adore math or regard it as their least favorite. No other core subject has the same infamy as math for generating passionate reactions both for and against it. It has applications in every field, whether basic operations or complex calculus problems. Knowing the basics of math is necessary to do any work properly. 

If you are writing essays about Math, we have compiled some essay examples for you to get started. 

1. Mathematics: Problem Solving and Ideal Math Classroom by Darlene Gregory 

2. math essay by prasanna, 3. short essay on the importance of mathematics by jay prakash.

  • 4.  Math Anxiety by Elias Wong

5. Why Math Isn’t as Useless as We Think by Murtaza Ali

1. mathematics – do you love or hate it, 2. why do many people despise math, 3. how does math prepare you for the future, 4. is mathematics an essential skill, 5. mathematics in the modern world.

“The trait of the teacher that is being strict is we know that will really help the students to change. But it will give a stress and pressure to students and that is one of the causes why students begin to dislike math. As a student I want a teacher that is not so much strict and giving considerations to his students. A teacher that is not giving loads of things to do and must know how to understand the reasons of his students.”

Gregory discusses the reasons for most students’ hatred of math and how teachers handle the subject in class. She says that math teachers do not explain the topics well, give too much work, and demand nothing less than perfection. To her, the ideal math class would involve teachers being more considerate and giving less work. 

You might also be interested in our ordinal number explainer.

“Math is complicated to learn, and one needs to focus and concentrate more. Math is logical sometimes, and the logic needs to be derived out. Maths make our life easier and more straightforward. Math is considered to be challenging because it consists of many formulas that have to be learned, and many symbols and each symbol generally has its significance.”

In her essay, Prasanna gives readers a basic idea of what math is and its importance. She additionally lists down some of the many uses of mathematics in different career paths, namely managing finances, cooking, home modeling and construction, and traveling. Math may seem “useless” and “annoying” to many, but the essay gives readers a clear message: we need math to succeed. 

“In this modern age of Science and Technology, emphasis is given on Science such as Physics, Chemistry, Biology, Medicine and Engineering. Mathematics, which is a Science by any criterion, also is an efficient and necessary tool being employed by all these Sciences. As a matter of fact, all these Sciences progress only with the aid of Mathematics. So it is aptly remarked, ‘Mathematics is a Science of all Sciences and art of all arts.’”

As its title suggests, Prakash’s essay briefly explains why math is vital to human nature. As the world continues to advance and modernize, society emphasizes sciences such as medicine, chemistry, and physics. All sciences employ math; it cannot be studied without math. It also helps us better our reasoning skills and maximizes the human mind. It is not only necessary but beneficial to our everyday lives. 

4.   Math Anxiety by Elias Wong

“Math anxiety affects different not only students but also people in different ways. It’s important to be familiar with the thoughts you have about yourself and the situation when you encounter math. If you are aware of unrealistic or irrational thoughts you can work to replace those thoughts with more positive and realistic ones.”

Wong writes about the phenomenon known as “math anxiety.” This term is used to describe many people’s hatred or fear of math- they feel that they are incapable of doing it. This anxiety is caused mainly by students’ negative experiences in math class, which makes them believe they cannot do well. Wong explains that some people have brains geared towards math and others do not, but this should not stop people from trying to overcome their math anxiety. Through review and practice of basic mathematical skills, students can overcome them and even excel at math. 

“We see that math is not an obscure subject reserved for some pretentious intellectual nobility. Though we may not be aware of it, mathematics is embedded into many different aspects of our lives and our world — and by understanding it deeply, we may just gain a greater understanding of ourselves.”

Similar to some of the previous essays, Ali’s essay explains the importance of math. Interestingly, he tells a story of the life of a person name Kyle. He goes through the typical stages of life and enjoys typical human hobbies, including Rubik’s cube solving. Throughout this “Kyle’s” entire life, he performed the role of a mathematician in various ways. Ali explains that math is much more prevalent in our lives than we think, and by understanding it, we can better understand ourselves. 

Writing Prompts on Essays about Math

Math is a controversial subject that many people either passionately adore or despise. In this essay, reflect on your feelings towards math, and state your position on the topic. Then, give insights and reasons as to why you feel this way. Perhaps this subject comes easily to you, or perhaps it’s a subject that you find pretty challenging. For an insightful and compelling essay, you can include personal anecdotes to relate to your argument. 

Essays about Math: Why do many people despise math?

It is well-known that many people despise math. In this essay, discuss why so many people do not enjoy maths and struggle with this subject in school. For a compelling essay, gather interview data and statistics to support your arguments. You could include different sections correlating to why people do not enjoy this subject.

In this essay, begin by reading articles and essays about the importance of studying math. Then, write about the different ways that having proficient math skills can help you later in life. Next, use real-life examples of where maths is necessary, such as banking, shopping, planning holidays, and more! For an engaging essay, use some anecdotes from your experiences of using math in your daily life.

Many people have said that math is essential for the future and that you shouldn’t take a math class for granted. However, many also say that only a basic understanding of math is essential; the rest depends on one’s career. Is it essential to learn calculus and trigonometry? Choose your position and back up your claim with evidence. 

Prasanna’s essay lists down just a few applications math has in our daily lives. For this essay, you can choose any activity, whether running, painting, or playing video games, and explain how math is used there. Then, write about mathematical concepts related to your chosen activity and explain how they are used. Finally, be sure to link it back to the importance of math, as this is essentially the topic around which your essay is based. 

If you are interested in learning more, check out our essay writing tips !

For help with your essays, check out our round-up of the best essay checkers

math skills essay

Martin is an avid writer specializing in editing and proofreading. He also enjoys literary analysis and writing about food and travel.

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Essay: Why Is Math Important And How To Pass It?

  • Essay: Why Is Math Important…

Why is math important? By now, it is obvious to see that computers are taking our jobs. In a few years, robots will take more than 15% of our current jobs. It teaches us about the importance of math. So why is math important? It’s not too complicated. We use math everyday. For example, we use math when crossing the road, cooking, telling the time, and many more.

It’s possible to live our lives with basic addition, subtraction, and multiplication. Moreover, computers can do more than simple arithmetic. Engineers created the technologies that are taking our jobs. Math is important because if we can’t do what computers can’t do, then we can kiss our jobs goodbye. We can fix it by learning higher math. Students must come together, and we must decide why math is important for our lives.

As college students, we’re expected to have mastered high school math standards. Moreover, some of us had mathematical difficulties in high school.

That’s not a problem because we’re teachable. Brain scan has shown us that the brain can be regrown. It can be regrown through games, food, reading etc. Some of us enrolled in degrees such as nursing, art, math, etc.

All of us in this degree are expected to take math. We all have different reasons for enrolling in math requirement courses, so there’s no need to drop our degrees. It is necessary to understand that everyone has trouble in math, and our teachers had problems in math.

Therefore, we can improve our weaknesses in math by knowing our technological aid, our educational background, and our classroom environments.

Firstly, technological aid are technologies that assist students to learn and help teachers to teach easier. This includes calculators, math multimedia programs, and electronic boards, etc. Mymathlab is a popular technological aid at our school. It’s excellent because it educates students both in class and at home.

It is a sufficient condition because it takes us through the hard problems and provides answers with suitable explanations. According to Abbas Johari, “inductive multimedia programs help students understand math better”. Mr. Johari expanded on a study of multimedia programs using graph and word problems to educate 98 undergraduate students of a large southwest college who were enrolled in a computer literacy course. The research was successful. Because by the end of the research, students showed high improvement in their grades.

The problem is that MyMathLab prevails in spatial learning than in kinesthetic (hands-on learning), auditory (Listening learning), and linguistic(language learning). However, some of us learn differently, and some of us abuse the program because we copy and paste answers. Some researchers believe that students’ performance in math depends on the mathematics curriculum.

According to John K Alsup and Mark J. Sprigler, the ” traditional method(Houghton-Mifflin) showed positive results than the reform method (Cord Applied). The traditional method is based on non-reality problems, but the reform method is based on real problems. It depends on the teachers who teach it. There were about 335 eighth-graders in western United State from different areas that participated in this research. They did it by comparing SAT results of students from both curriculums.

It makes sense because schools such as Harvard University, M.I.T, and UMUC use different math curriculums. Because if schools used the same math curriculum, it wouldn’t make sense to pay high tuition for Harvard. Instead, we can pay less for UMUC and achieve equivalent knowledge.

Mymathlab is part of our curriculum that is given to us by our schools. If we know the program isn’t sufficient, then we should tell the school board. We need to take responsibility for our failures and do better.

Secondly, Educational background refers to our family’s educational values, our learning styles, and our understanding of math. If we know our educational background, then it prepares us for our careers. It is not late to prepare our mindset. Because it reveals our strengths and our weaknesses in math, it gives us the opportunity to grow our weaknesses.

It is a necessary condition that helps us keep track of the lesson. If our parents have a high educational value, then they can help us in math. It helps us know our learning styles. It facilitates learning to the best of our understanding, and it will lead us to do better in math. Knowing all of this can exponentially increase our understanding and performance in mathematics.

Moreover, some of us are doing well without the knowledge of our educational background. So why does it matter?  Some of us have tried it so many times and gave up on it. It is not surprising, math is hard. Some of us have math learning disabilities. The ability of low understanding in math. According to Emmanuel Manalo, Julie K. Bunnell, and Jennifer A. Stillman, “Students with math learning disabilities can improve highly through process mnemonics”.

They conducted a study with 13-14-year old’s, and they were distributed to various groups. Two Experiments were conducted to test Process mnemonics, no instruction, and demonstration imitation. They used different variables. Process mnemonics are using techniques such as rhymes to remember things. Demonstration imitation is the modeling of what you see.

We all know it’s a challenging subject. Some of us have come far without any knowledge of our educational background; If we attain it, then we’re guaranteed to improve our current educational values in math by 5%. Some of us with learning disabilities can improve. According to John Woodward, ” Strategies of teaching facts and extensive practice drills can help develop automaticity in math”.

This study was conducted on 58 fourth graders. Some with learning disabilities, and some with no disabilities. It proves that practice drills and strategies that teachers use to teach us can improve our math understanding.

For example, the speech teacher asked Brandon and his classmates if they knew their learning styles. Since Brandon knew his learning style because his DVR-0061 teacher taught him. He was among the students who knew what she was talking about.  It shows how important learning styles are.

Thirdly, the classroom environment is an important aspect of the understanding. The classroom environment includes climate, teachers, and classmates. Climate is an important fact in understanding math. Because if students are not comfortable in their environment, then they will have a hard time perceiving and processing mathematical information.

For instance, if it is hot outside and a student goes to class. He/she will expect the classroom to be cold. If the temperature in the classroom is the same as the temperature outside, then the student will have a hard time learning. Sometimes, we fail to ask our teachers questions when we don’t understand. It’s important to have a teacher that cares about math.

If teachers don’t care, then students won’t care. We don’t take into consideration that our classmates can tutor us. At times, we understand better from classmates than from our teachers.

According to Karen J. Graham and Francis Fennell, ” Successful teaching depends on teachers’ ability to make decisions based on their knowledge of mathematics, the curriculum expectations, the classroom/ school environment, and the needs of the students”. Our teachers are part of our misunderstanding. If so, why are our teachers allowed to teach us?

Because teachers are meant to teach us depending on our understanding, they are supposed to boost our self-esteem and self-concept in math. Furthermore, teachers are also responsible for everything in the classroom. It implies that teachers need to fix the temperature if it doesn’t correlate with the students in the class.

Is it right to blame our teachers for our failures and misunderstanding? If we think about it as college students, it’s our responsibility to fix our classroom environmental issues. First, we must suggest our opinions with our teachers on class environmental changes. If our teachers can’t help, then we can dress depending on the classroom temperature.

If we don’t understand, then it’s our responsibility to go and find a tutor. For instance, our schools provide us with free tutoring for every subject available. It s up to us to carry ourselves there to learn.

Improvement in mathematical skills is possible through our technological aid, educational background, and classroom environments. Mathematics is important in many ways. First, it is the foundation of the world. Because currently, robots are already taking our jobs, no one will want to employ a person if a computer is faster and better than them. Plus, they don’t need to pay for a computer. Many majors we take today require math, and we need math in our daily lives.

Computers are structured using math, but their math skills are limited. Math is a hard subject, but we can understand it by managing our tech-aides, learning background, and environments. We need to take advantage of the resources around us. Especially us in community colleges because the world does not care about who does the job.

They only want people who can get the job done and fast. We must keep practicing until we understand. Therefore, we must not give up on math under any circumstances because we have dreams to conquer.

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Tutor and Freelance Writer. Science Teacher and Lover of Essays. Article last reviewed: 2022 | St. Rosemary Institution © 2010-2023 | Creative Commons 4.0

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9 Ways to Improve Math Skills Quickly & Effectively

Overhead view of a child using a piece of paper, a pen, and a calculator to do math homework and improve their math skills

Written by Ashley Crowe

Help your child improve their math skills with the game that makes learning an adventure!

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The importance of understanding basic math skills

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Math class can move pretty fast. There’s so much to cover in the course of a school year. And if your child doesn’t get a new math idea right away, they can quickly get left behind.

If your child is struggling with basic math problems every day, it doesn’t mean they’re destined to be bad at math. Some students need more time to develop the problem-solving skills that math requires. Others may need to revisit past concepts before moving on. Because of how math is structured, it’s best to take each year step-by-step, lesson by lesson.

This article has tips and tricks to improve your child’s math skills while minimizing frustrations and struggles. If your child is growing to hate math, read on for ways to improve their skills and confidence, and maybe even make math fun! 

But first, the basics.

Math is a subject that builds on itself. It takes a solid understanding of past concepts to prepare for the next lesson. 

That’s why math can become frustrating when you’re forced to move on before you’re ready. You’re either stuck trying to catch up or you end up falling further behind.

But with a strong understanding of basic math skills, your child can be set up for school success. If you’re unfamiliar with the idea of sets or whole numbers , this is a great place to start. 

What are considered basic math skills?

The basic math skills required to move on to higher levels of math learning are: 

  • Addition — Adding to a set.
  • Subtraction — Taking away from a set.
  • Multiplication — Adding equal sets together in groups (2 sets of 3 is the same as 2x3, or 6).
  • Division — How many equal sets can be found in a number (12 has how many sets of two in it? 6 sets of 2).
  • Percentages — A specific amount in relation to 100.
  • Fractions & Decimals — Fractions are equal parts of a whole set. Decimals represent a number of parts of a whole in relation to 10. These both contrast with whole numbers. 
  • Spatial Reasoning — How numbers and shapes fit together.

How to improve math skills 

People aren’t bad at math — many just need more time and practice to gain a thorough understanding.

How can you help your child improve their math abilities? Use our top 9 tips for quickly and effectively improving math skills .

1. Wrap your head around the concepts

Repetition and practice are great, but if you don’t understand the concept , it will be difficult to move forward. 

Luckily, there are many great ways to break down math concepts . The trick is finding the one that works best for your child.

Math manipulatives can be a game-changer for children who are struggling with big math ideas. Taking math off the page and putting it into their hands can bring ideas to life. Numbers become less abstract and more concrete when you’re counting toy cars or playing with blocks. Creating these “sets” of objects can bring clarity to basic math learning.

2. Try game-based learning

During math practice, repetition is important — but it can get old in a hurry. No one enjoys copying their times tables over and over and over again. If learning math has become a chore, it’s time to bring back the fun! 

Game-based learning is a great way to practice new concepts and solidify past lessons. It can even make repetition fun and engaging.

Game-based learning can look like a family board game on Friday night or an educational app , like Prodigy Math .

A glimpse of the Prodigy Math Game world and a sample math question a kid could receive to help improve their math skills while playing.

Take math from frustrating to fun with the right game, then watch the learning happen easily!

3. Bring math into daily life

You use basic math every day. 

As you go about your day, help your child see the math that’s all around them:  

  • Tell them how fast you’re driving on the way to school
  • Calculate the discount you’ll receive on your next Target trip
  • Count out the number of apples you need to buy at the grocery store
  • While baking, explain how 6 quarter cups is the same amount of flour as a cup and a half — then enjoy some cookies!

Relate math back to what your child loves and show them how it’s used every day. Math doesn’t have to be mysterious or abstract. Instead, use math to race monster trucks or arrange tea parties. Break it down, take away the fear, and watch their interest in math grow.

4. Implement daily practice

Math practice is important. Once you understand the concept, you have to nail down the mechanics. And often, it’s the practice that finally helps the concept click. Either way, math requires more than just reading formulas on a page.

Daily practice can be tough to implement, especially with a math-averse child. This is a great time to bring out the game-based learning mentioned above. Or find an activity that lines up with their current lesson. Are they learning about squares? Break out the math link cubes and create them. Whenever possible, step away from the worksheets and flashcards and find practice elsewhere.

5. Sketch word problems

Nothing causes a panic quite like an unexpected word problem. Something about the combination of numbers and words can cause the brain of a struggling math learner to shut down. But it doesn’t have to be that way.

Many word problems just need to be broken down, step by step . One great way to do this is to sketch it out. If Doug has five apples and four oranges, then eats two of each, how many does he have left? Draw it, talk it out, cross them off, then count. 

If you’ve been talking your child through the various math challenges you encounter every day, many word problems will start to feel familiar. 

6. Set realistic goals

If your child has fallen behind in math, then more study time is the answer. But forcing them to cram an extra hour of math in their day is not likely to produce better results. To see a positive change, first identify their biggest struggles . Then set realistic goals addressing these issues . 

Two more hours of practicing a concept they don’t understand is only going to cause more frustration. Even if they can work through the mechanics of a problem, the next lesson will leave them feeling just as lost. 

Instead, try mini practice sessions and enlist some extra help. Approach the problem in a new way, reach out to their teacher or try an online math lesson . Make sure the extra time is troubleshooting the actual problem, not just reinforcing the idea that math is hard and no fun. 

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Set Goals and Rewards in Prodigy Math

Did you know that parents can set learning goals for their child in Prodigy Math? And once they achieve them, they'll unlock in-game rewards of your choice!

7. Engage with a math tutor

If your child is struggling with big picture concepts, look into finding a math tutor . Everyone learns differently, and you and your child’s teacher may be missing that “aha” moment that a little extra time and the right tutor can provide.

It’s amazing when a piece of the math puzzle finally clicks for your child. If you’re ready to get that extra help, try a free 1:1 online session from Prodigy Math Tutoring. Prodigy’s tutors are real teachers who know how to connect kids to math. With the right approach, your child can become confident in math — and who knows, they may even begin to enjoy it. 

8. Focus on one concept at a time

Math builds on itself. If your child is struggling through their current lesson, they can’t skip it and come back to it later. This is the time to practice and repeat — re-examining and reinforcing the current concept until it makes sense.

Look for other ways to approach new math ideas. Use math manipulatives to bring numbers off the page. Or try a learning app with exciting rewards and positive reinforcement to encourage extra practice. 

Take a step back when frustrations get high — but resist the temptation to just let it go. Once the concept clicks, they’ll be excited to forge ahead.

9. Teach others math you already know

Even if your child is struggling in math, they’ve still learned so much since last year. Focus on the improvements they’ve made and let them showcase their knowledge. If they have younger siblings, your older child can demonstrate addition or show them how to use a number line. This is a great way to build their confidence and encourage them to keep going.

Or let them teach you how they solve new problems. Have your child talk you through the process while you solve a long division problem . You’re likely to find yourself a little rusty on the details. Play it up and get a little silly. They’ll love teaching you the ropes of this “new math.”

Child using movable numbers and math symbols on a table to show a 5x5 formula and help someone else improve their math skills

Embracing technology to improve math skills

Though much of your math learning was done with pencil to paper, there are many more ways to build number skills in today’s tech world. 

Your child can take live, online math courses to work through tough concepts. Or play a variety of online games, solving math puzzles and getting consistent practice while having fun.

These technical advances can help every child learn math, no matter their preferred learning or study style. If your child is a visual learner, there’s an app for that. Do they process best while working in groups? Jump online and find one. Don’t keep repeating the same lessons from their math class over and over. Branch out, try something new and watch the learning click. 

Look online for more math help

There are so many online resources, it can be hard to know where to start. 

At Prodigy, we’re happy to help you get the ball rolling on your child’s math learning, from kindergarten through 8th grade. It’s free to sign up, fun to play and exciting to watch as your child’s math understanding grows.

Sign up for a free parent account and get instant data on your child’s progress as they build more math skills with Prodigy Math Game . It’s time to take the math struggle out of your home and enjoy learning together!

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How to Improve Math Skills

Last Updated: January 12, 2023 Approved

This article was co-authored by Daron Cam and by wikiHow staff writer, Hannah Madden . Daron Cam is an Academic Tutor and the Founder of Bay Area Tutors, Inc., a San Francisco Bay Area-based tutoring service that provides tutoring in mathematics, science, and overall academic confidence building. Daron has over eight years of teaching math in classrooms and over nine years of one-on-one tutoring experience. He teaches all levels of math including calculus, pre-algebra, algebra I, geometry, and SAT/ACT math prep. Daron holds a BA from the University of California, Berkeley and a math teaching credential from St. Mary's College. wikiHow marks an article as reader-approved once it receives enough positive feedback. This article received 28 testimonials and 84% of readers who voted found it helpful, earning it our reader-approved status. This article has been viewed 405,978 times.

There’s no doubt about it: math is tough. As a result, a lot of kids (and adults!) struggle with math at some point in their lives. By building up your skills and practicing every day, you can make math a little less frustrating and have a higher chance of success. Use these tips and tricks during school, while you’re studying, and when you’re out and about to break down and complete math problems easily.

Play math games.

Build up your skills while having some fun!

  • DragonBox 5+ which lets you gradually build your algebra skills until you’re able to master more and more advanced equations.
  • Prodigy, a game targeted at elementary-school students, that integrates math practice into a role-playing game that allows players to use math to make their way through an appealing fantasy world.
  • Polyup, a calculator-based math game for more advanced high school and college students.

Practice math in everyday scenarios.

Make math part of your daily life to practice it without even thinking.

  • Or, if you plan to hike a new trail that’s 7 miles long and it takes you 20 minutes to walk a mile, how long should you plan for your hike to take? (2 hours and 36 minutes).

Use mental math if you can.

Doing math in your head can help you remember key concepts.

  • If you’re worried about your mental math skills, you can always double check your answer on your phone or computer.

Review math concepts every day.

Practice makes perfect, and math is no exception.

  • Make note cards. Write out important concepts and formulas on note cards so that you can easily refer to them while doing problems and use them for study guides before exams.
  • Study in a quiet place. Distractions, whether aural or visual, will detract both from your ability to pay attention and to retain information.
  • Study when you’re alert and rested. Don’t try to force yourself to study late at night or when you’re sleep-deprived.

Show your work, not just your answers.

Writing it all out can help you spot mistakes.

  • Showing your work can also help you check your answers on homework and test problems.
  • Don’t solve math problems with a pen! Use a pencil so you can erase and correct mistakes if they happen.

Sketch out word problems to give yourself a visual.

Word problems are usually tougher than straight math problems.

  • For example, a problem might say, “If you have 4 pieces of candy split evenly into 2 bags, how many pieces of candy are in each bag?” You could draw 2 squares to represent the bags, then fill in 4 circles split between them to represent the candy.

Practice with example problems.

You can find tons of example problems online.

  • Your teacher might also be able to give you some extra example problems if you ask for them.
  • Using example problems is a great way to practice for a test.

Look up lessons online.

Sometimes you need a little more explanation to really understand.

  • PatrickJMT on YouTube, a college math professor
  • Khan Academy, a website with video lessons and interactive study guides
  • Breaking Math, a podcast for math concepts

Master one concept before moving onto the next.

In math, each topic builds upon the last one.

Teach math problem or concept to someone else.

Learning by teaching someone is a great way to solidify concepts.

  • Have your friend or family member ask you questions, too. Try to answer them as best you can to really practice.

Expert Q&A

Daron Cam

  • Try not to fall behind in your homework or schoolwork. The more you keep up in class, the easier it will be. Thanks Helpful 4 Not Helpful 0

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Improve Your Math Grade

  • ↑ Daron Cam. Academic Tutor. Expert Interview. 29 May 2020.
  • ↑ http://www.schoolfamily.com/school-family-articles/article/10785-mastering-math
  • ↑ https://www.edutopia.org/article/5-tips-improving-students-success-math
  • ↑ https://math.osu.edu/undergrad/non-majors/resources/study-math-college
  • ↑ https://www.youtube.com/watch?t=96&v=aIRh_15O2S0&feature=youtu.be
  • ↑ https://www.mathgoodies.com/articles/improve_your_grades

About This Article

Daron Cam

To improve your math skills, start by taking good notes in class and asking lots of questions to understand the material. Then, schedule time each day to study from your notes and do your homework. When you study, do practice problems to cement your comprehension of the math. In addition to studying, try playing math games online, such as DragonBox 5+ or Prodigy, which will help hone your math skills in a fun way. For ways to incorporate math into your everyday life, read on! Did this summary help you? Yes No

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National Academies Press: OpenBook

High School Mathematics at Work: Essays and Examples for the Education of All Students (1998)

Chapter: part one: connecting mathematics with work and life, part one— connecting mathematics with work and life.

Mathematics is the key to opportunity. No longer just the language of science, mathematics now contributes in direct and fundamental ways to business, finance, health, and defense. For students, it opens doors to careers. For citizens, it enables informed decisions. For nations, it provides knowledge to compete in a technological community. To participate fully in the world of the future, America must tap the power of mathematics. (NRC, 1989, p. 1)

The above statement remains true today, although it was written almost ten years ago in the Mathematical Sciences Education Board's (MSEB) report Everybody Counts (NRC, 1989). In envisioning a future in which all students will be afforded such opportunities, the MSEB acknowledges the crucial role played by formulae and algorithms, and suggests that algorithmic skills are more flexible, powerful, and enduring when they come from a place of meaning and understanding. This volume takes as a premise that all students can develop mathematical understanding by working with mathematical tasks from workplace and everyday contexts . The essays in this report provide some rationale for this premise and discuss some of the issues and questions that follow. The tasks in this report illuminate some of the possibilities provided by the workplace and everyday life.

Contexts from within mathematics also can be powerful sites for the development of mathematical understanding, as professional and amateur mathematicians will attest. There are many good sources of compelling problems from within mathematics, and a broad mathematics education will include experience with problems from contexts both within and outside mathematics. The inclusion of tasks in this volume is intended to highlight particularly compelling problems whose context lies outside of mathematics, not to suggest a curriculum.

The operative word in the above premise is "can." The understandings that students develop from any encounter with mathematics depend not only on the context, but also on the students' prior experience and skills, their ways of thinking, their engagement with the task, the environment in which they explore the task—including the teacher, the students, and the tools—the kinds of interactions that occur in that environment, and the system of internal and external incentives that might be associated with the activity. Teaching and learning are complex activities that depend upon evolving and rarely articulated interrelationships among teachers, students, materials, and ideas. No prescription for their improvement can be simple.

This volume may be beneficially seen as a rearticulation and elaboration of a principle put forward in Reshaping School Mathematics :

Principle 3: Relevant Applications Should be an Integral Part of the Curriculum.

Students need to experience mathematical ideas in the context in which they naturally arise—from simple counting and measurement to applications in business and science. Calculators and computers make it possible now to introduce realistic applications throughout the curriculum.

The significant criterion for the suitability of an application is whether it has the potential to engage students' interests and stimulate their mathematical thinking. (NRC, 1990, p. 38)

Mathematical problems can serve as a source of motivation for students if the problems engage students' interests and aspirations. Mathematical problems also can serve as sources of meaning and understanding if the problems stimulate students' thinking. Of course, a mathematical task that is meaningful to a student will provide more motivation than a task that does not make sense. The rationale behind the criterion above is that both meaning and motivation are required. The motivational benefits that can be provided by workplace and everyday problems are worth mentioning, for although some students are aware that certain mathematics courses are necessary in order to gain entry into particular career paths, many students are unaware of how particular topics or problem-solving approaches will have relevance in any workplace. The power of using workplace and everyday problems to teach mathematics lies not so much in motivation, however, for no con-

text by itself will motivate all students. The real power is in connecting to students' thinking.

There is growing evidence in the literature that problem-centered approaches—including mathematical contexts, "real world" contexts, or both—can promote learning of both skills and concepts. In one comparative study, for example, with a high school curriculum that included rich applied problem situations, students scored somewhat better than comparison students on algebraic procedures and significantly better on conceptual and problem-solving tasks (Schoen & Ziebarth, 1998). This finding was further verified through task-based interviews. Studies that show superior performance of students in problem-centered classrooms are not limited to high schools. Wood and Sellers (1996), for example, found similar results with second and third graders.

Research with adult learners seems to indicate that "variation of contexts (as well as the whole task approach) tends to encourage the development of general understanding in a way which concentrating on repeated routine applications of algorithms does not and cannot" (Strässer, Barr, Evans, & Wolf, 1991, p. 163). This conclusion is consistent with the notion that using a variety of contexts can increase the chance that students can show what they know. By increasing the number of potential links to the diverse knowledge and experience of the students, more students have opportunities to excel, which is to say that the above premise can promote equity in mathematics education.

There is also evidence that learning mathematics through applications can lead to exceptional achievement. For example, with a curriculum that emphasizes modeling and applications, high school students at the North Carolina School of Science and Mathematics have repeatedly submitted winning papers in the annual college competition, Mathematical Contest in Modeling (Cronin, 1988; Miller, 1995).

The relationships among teachers, students, curricular materials, and pedagogical approaches are complex. Nonetheless, the literature does supports the premise that workplace and everyday problems can enhance mathematical learning, and suggests that if students engage in mathematical thinking, they will be afforded opportunities for building connections, and therefore meaning and understanding.

In the opening essay, Dale Parnell argues that traditional teaching has been missing opportunities for connections: between subject-matter and context, between academic and vocational education, between school and life, between knowledge and application, and between subject-matter disciplines. He suggests that teaching must change if more students are to learn mathematics. The question, then, is how to exploit opportunities for connections between high school mathematics and the workplace and everyday life.

Rol Fessenden shows by example the importance of mathematics in business, specifically in making marketing decisions. His essay opens with a dialogue among employees of a company that intends to expand its business into

Japan, and then goes on to point out many of the uses of mathematics, data collection, analysis, and non-mathematical judgment that are required in making such business decisions.

In his essay, Thomas Bailey suggests that vocational and academic education both might benefit from integration, and cites several trends to support this suggestion: change and uncertainty in the workplace, an increased need for workers to understand the conceptual foundations of key academic subjects, and a trend in pedagogy toward collaborative, open-ended projects. Further-more, he observes that School-to-Work experiences, first intended for students who were not planning to attend a four-year college, are increasingly being seen as useful in preparing students for such colleges. He discusses several such programs that use work-related applications to teach academic skills and to prepare students for college. Integration of academic and vocational education, he argues, can serve the dual goals of "grounding academic standards in the realistic context of workplace requirements and introducing a broader view of the potential usefulness of academic skills even for entry level workers."

Noting the importance and utility of mathematics for jobs in science, health, and business, Jean Taylor argues for continued emphasis in high school of topics such as algebra, estimation, and trigonometry. She suggests that workplace and everyday problems can be useful ways of teaching these ideas for all students.

There are too many different kinds of workplaces to represent even most of them in the classrooms. Furthermore, solving mathematics problems from some workplace contexts requires more contextual knowledge than is reasonable when the goal is to learn mathematics. (Solving some other workplace problems requires more mathematical knowledge than is reasonable in high school.) Thus, contexts must be chosen carefully for their opportunities for sense making. But for students who have knowledge of a workplace, there are opportunities for mathematical connections as well. In their essay, Daniel Chazan and Sandra Callis Bethell describe an approach that creates such opportunities for students in an algebra course for 10th through 12th graders, many of whom carried with them a "heavy burden of negative experiences" about mathematics. Because the traditional Algebra I curriculum had been extremely unsuccessful with these students, Chazan and Bethell chose to do something different. One goal was to help students see mathematics in the world around them. With the help of community sponsors, Chazen and Bethell asked students to look for mathematics in the workplace and then describe that mathematics and its applications to their classmates.

The tasks in Part One complement the points made in the essays by making direct connections to the workplace and everyday life. Emergency Calls (p. 42) illustrates some possibilities for data analysis and representation by discussing the response times of two ambulance companies. Back-of-the-Envelope Estimates (p. 45) shows how quick, rough estimates and calculations

are useful for making business decisions. Scheduling Elevators (p. 49) shows how a few simplifying assumptions and some careful reasoning can be brought together to understand the difficult problem of optimally scheduling elevators in a large office building. Finally, in the context of a discussion with a client of an energy consulting firm, Heating-Degree-Days (p. 54) illuminates the mathematics behind a common model of energy consumption in home heating.

Cronin, T. P. (1988). High school students win "college" competition. Consortium: The Newsletter of the Consortium for Mathematics and Its Applications , 26 , 3, 12.

Miller, D. E. (1995). North Carolina sweeps MCM '94. SIAM News , 28 (2).

National Research Council. (1989). Everybody counts: A report to the nation on the future of mathematics education . Washington, DC: National Academy Press.

National Research Council. (1990). Reshaping school mathematics: A philosophy and framework for curriculum . Washington, DC: National Academy Press.

Schoen, H. L. & Ziebarth, S. W. (1998). Assessment of students' mathematical performance (A Core-Plus Mathematics Project Field Test Progress Report). Iowa City: Core Plus Mathematics Project Evaluation Site, University of Iowa.

Strässer, R., Barr, G. Evans, J. & Wolf, A. (1991). Skills versus understanding. In M. Harris (Ed.), Schools, mathematics, and work (pp. 158-168). London: The Falmer Press.

Wood, T. & Sellers, P. (1996). Assessment of a problem-centered mathematics program: Third grade. Journal for Research in Mathematics Education , 27 (3), 337-353.

1— Mathematics as a Gateway to Student Success

DALE PARNELL

Oregon State University

The study of mathematics stands, in many ways, as a gateway to student success in education. This is becoming particularly true as our society moves inexorably into the technological age. Therefore, it is vital that more students develop higher levels of competency in mathematics. 1

The standards and expectations for students must be high, but that is only half of the equation. The more important half is the development of teaching techniques and methods that will help all students (rather than just some students) reach those higher expectations and standards. This will require some changes in how mathematics is taught.

Effective education must give clear focus to connecting real life context with subject-matter content for the student, and this requires a more ''connected" mathematics program. In many of today's classrooms, especially in secondary school and college, teaching is a matter of putting students in classrooms marked "English," "history," or "mathematics," and then attempting to fill their heads with facts through lectures, textbooks, and the like. Aside from an occasional lab, workbook, or "story problem," the element of contextual teaching and learning is absent, and little attempt is made to connect what students are learning with the world in which they will be expected to work and spend their lives. Often the frag-

mented information offered to students is of little use or application except to pass a test.

What we do in most traditional classrooms is require students to commit bits of knowledge to memory in isolation from any practical application—to simply take our word that they "might need it later." For many students, "later" never arrives. This might well be called the freezer approach to teaching and learning. In effect, we are handing out information to our students and saying, "Just put this in your mental freezer; you can thaw it out later should you need it." With the exception of a minority of students who do well in mastering abstractions with little contextual experience, students aren't buying that offer. The neglected majority of students see little personal meaning in what they are asked to learn, and they just don't learn it.

I recently had occasion to interview 75 students representing seven different high schools in the Northwest. In nearly all cases, the students were juniors identified as vocational or general education students. The comment of one student stands out as representative of what most of these students told me in one way or another: "I know it's up to me to get an education, but a lot of times school is just so dull and boring. … You go to this class, go to that class, study a little of this and a little of that, and nothing connects. … I would like to really understand and know the application for what I am learning." Time and again, students were asking, "Why do I have to learn this?" with few sensible answers coming from the teachers.

My own long experience as a community college president confirms the thoughts of these students. In most community colleges today, one-third to one-half of the entering students are enrolled in developmental (remedial) education, trying to make up for what they did not learn in earlier education experiences. A large majority of these students come to the community college with limited mathematical skills and abilities that hardly go beyond adding, subtracting, and multiplying with whole numbers. In addition, the need for remediation is also experienced, in varying degrees, at four-year colleges and universities.

What is the greatest sin committed in the teaching of mathematics today? It is the failure to help students use the magnificent power of the brain to make connections between the following:

  • subject-matter content and the context of use;
  • academic and vocational education;
  • school and other life experiences;
  • knowledge and application of knowledge; and
  • one subject-matter discipline and another.

Why is such failure so critical? Because understanding the idea of making the connection between subject-matter content and the context of application

is what students, at all levels of education, desperately require to survive and succeed in our high-speed, high-challenge, rapidly changing world.

Educational policy makers and leaders can issue reams of position papers on longer school days and years, site-based management, more achievement tests and better assessment practices, and other "hot" topics of the moment, but such papers alone will not make the crucial difference in what students know and can do. The difference will be made when classroom teachers begin to connect learning with real-life experiences in new, applied ways, and when education reformers begin to focus upon learning for meaning.

A student may memorize formulas for determining surface area and measuring angles and use those formulas correctly on a test, thereby achieving the behavioral objectives set by the teacher. But when confronted with the need to construct a building or repair a car, the same student may well be left at sea because he or she hasn't made the connection between the formulas and their real-life application. When students are asked to consider the Pythagorean Theorem, why not make the lesson active, where students actually lay out the foundation for a small building like a storage shed?

What a difference mathematics instruction could make for students if it were to stress the context of application—as well as the content of knowledge—using the problem-solving model over the freezer model. Teaching conducted upon the connected model would help more students learn with their thinking brain, as well as with their memory brain, developing the competencies and tools they need to survive and succeed in our complex, interconnected society.

One step toward this goal is to develop mathematical tasks that integrate subject-matter content with the context of application and that are aimed at preparing individuals for the world of work as well as for post-secondary education. Since many mathematics teachers have had limited workplace experience, they need many good examples of how knowledge of mathematics can be applied to real life situations. The trick in developing mathematical tasks for use in classrooms will be to keep the tasks connected to real life situations that the student will recognize. The tasks should not be just a contrived exercise but should stay as close to solving common problems as possible.

As an example, why not ask students to compute the cost of 12 years of schooling in a public school? It is a sad irony that after 12 years of schooling most students who attend the public schools have no idea of the cost of their schooling or how their education was financed. No wonder that some public schools have difficulty gaining financial support! The individuals being served by the schools have never been exposed to the real life context of who pays for the schools and why. Somewhere along the line in the teaching of mathematics, this real life learning opportunity has been missed, along with many other similar contextual examples.

The mathematical tasks in High School Mathematics at Work provide students (and teachers) with a plethora of real life mathematics problems and

challenges to be faced in everyday life and work. The challenge for teachers will be to develop these tasks so they relate as close as possible to where students live and work every day.

Parnell, D. (1985). The neglected majority . Washington, DC: Community College Press.

Parnell, D. (1995). Why do I have to learn this ? Waco, TX: CORD Communications.

D ALE P ARNELL is Professor Emeritus of the School of Education at Oregon State University. He has served as a University Professor, College President, and for ten years as the President and Chief Executive Officer of the American Association of Community Colleges. He has served as a consultant to the National Science Foundation and has served on many national commissions, such as the Secretary of Labor's Commission on Achieving Necessary Skills (SCANS). He is the author of the book The Neglected Majority which provided the foundation for the federally-funded Tech Prep Associate Degree Program.

2— Market Launch

ROL FESSENDEN

L. L. Bean, Inc.

"OK, the agenda of the meeting is to review the status of our launch into Japan. You can see the topics and presenters on the list in front of you. Gregg, can you kick it off with a strategy review?"

"Happy to, Bob. We have assessed the possibilities, costs, and return on investment of opening up both store and catalog businesses in other countries. Early research has shown that both Japan and Germany are good candidates. Specifically, data show high preference for good quality merchandise, and a higher-than-average propensity for an active outdoor lifestyle in both countries. Education, age, and income data are quite different from our target market in the U.S., but we do not believe that will be relevant because the cultures are so different. In addition, the Japanese data show that they have a high preference for things American, and, as you know, we are a classic American company. Name recognition for our company is 14%, far higher than any of our American competition in Japan. European competitors are virtually unrecognized, and other Far Eastern competitors are perceived to be of lower quality than us. The data on these issues are quite clear.

"Nevertheless, you must understand that there is a lot of judgment involved in the decision to focus on Japan. The analyses are limited because the cultures are different and we expect different behavioral drivers. Also,

much of the data we need in Japan are simply not available because the Japanese marketplace is less well developed than in the U.S. Drivers' license data, income data, lifestyle data, are all commonplace here and unavailable there. There is little prior penetration in either country by American retailers, so there is no experience we can draw upon. We have all heard how difficult it will be to open up sales operations in Japan, but recent sales trends among computer sellers and auto parts sales hint at an easing of the difficulties.

"The plan is to open three stores a year, 5,000 square feet each. We expect to do $700/square foot, which is more than double the experience of American retailers in the U.S. but 45% less than our stores. In addition, pricing will be 20% higher to offset the cost of land and buildings. Asset costs are approximately twice their rate in the U.S., but labor is slightly less. Benefits are more thoroughly covered by the government. Of course, there is a lot of uncertainty in the sales volumes we are planning. The pricing will cover some of the uncertainty but is still less than comparable quality goods already being offered in Japan.

"Let me shift over to the competition and tell you what we have learned. We have established long-term relationships with 500 to 1000 families in each country. This is comparable to our practice in the U.S. These families do not know they are working specifically with our company, as this would skew their reporting. They keep us appraised of their catalog and shopping experiences, regardless of the company they purchase from. The sample size is large enough to be significant, but, of course, you have to be careful about small differences.

"All the families receive our catalog and catalogs from several of our competitors. They match the lifestyle, income, and education demographic profiles of the people we want to have as customers. They are experienced catalog shoppers, and this will skew their feedback as compared to new catalog shoppers.

"One competitor is sending one 100-page catalog per quarter. The product line is quite narrow—200 products out of a domestic line of 3,000. They have selected items that are not likely to pose fit problems: primarily outerwear and knit shirts, not many pants, mostly men's goods, not women's. Their catalog copy is in Kanji, but the style is a bit stilted we are told, probably because it was written in English and translated, but we need to test this hypothesis. By contrast, we have simply mailed them the same catalog we use in the U.S., even written in English.

"Customer feedback has been quite clear. They prefer our broader assortment by a ratio of 3:1, even though they don't buy most of the products. As the competitors figured, sales are focused on outerwear and knits, but we are getting more sales, apparently because they like looking at the catalog and spend more time with it. Again, we need further testing. Another hypothesis is that our brand name is simply better known.

"Interestingly, they prefer our English-language version because they find it more of an adventure to read the catalog in another language. This is probably

a built-in bias of our sampling technique because we specifically selected people who speak English. We do not expect this trend to hold in a general mailing.

"The English language causes an 8% error rate in orders, but orders are 25% larger, and 4% more frequent. If we can get them to order by phone, we can correct the errors immediately during the call.

"The broader assortment, as I mentioned, is resulting in a significantly higher propensity to order, more units per order, and the same average unit cost. Of course, paper and postage costs increase as a consequence of the larger format catalog. On the other hand, there are production efficiencies from using the same version as the domestic catalog. Net impact, even factoring in the error rate, is a significant sales increase. On the other hand, most of the time, the errors cause us to ship the wrong item which then needs to be mailed back at our expense, creating an impression in the customers that we are not well organized even though the original error was theirs.

"Final point: The larger catalog is being kept by the customer an average of 70 days, while the smaller format is only kept on average for 40 days. Assuming—we need to test this—that the length of time they keep the catalog is proportional to sales volumes, this is good news. We need to assess the overall impact carefully, but it appears that there is a significant population for which an English-language version would be very profitable."

"Thanks, Gregg, good update. Jennifer, what do you have on customer research?"

"Bob, there's far more that we need to know than we have been able to find out. We have learned that Japan is very fad-driven in apparel tastes and fascinated by American goods. We expect sales initially to sky-rocket, then drop like a stone. Later on, demand will level out at a profitable level. The graphs on page 3 [ Figure 2-1 ] show demand by week for 104 weeks, and we have assessed several scenarios. They all show a good underlying business, but the uncertainty is in the initial take-off. The best data are based on the Italian fashion boom which Japan experienced in the late 80s. It is not strictly analogous because it revolved around dress apparel instead of our casual and weekend wear. It is, however, the best information available.

math skills essay

FIGURE 2-1: Sales projections by week, Scenario A

math skills essay

FIGURE 2-2: Size distributions, U.S. vs. Japan

"Our effectiveness in positioning inventory for that initial surge will be critical to our long-term success. There are excellent data—supplied by MITI, I might add—that show that Japanese customers can be intensely loyal to companies that meet their high service expectations. That is why we prepared several scenarios. Of course, if we position inventory for the high scenario, and we experience the low one, we will experience a significant loss due to liquidations. We are still analyzing the long-term impact, however. It may still be worthwhile to take the risk if the 2-year ROI 1 is sufficient.

"We have solid information on their size scales [ Figure 2-2 ]. Seventy percent are small and medium. By comparison, 70% of Americans are large and extra large. This will be a challenge to manage but will save a few bucks on fabric.

"We also know their color preferences, and they are very different than Americans. Our domestic customers are very diverse in their tastes, but 80% of Japanese customers will buy one or two colors out of an offering of 15. We are still researching color choices, but it varies greatly for pants versus shirts, and for men versus women. We are confident we can find patterns, but we also know that it is easy to guess wrong in that market. If we guess wrong, the liquidation costs will be very high.

"Bad news on the order-taking front, however. They don't like to order by phone. …"

In this very brief exchange among decision-makers we observe the use of many critically important skills that were originally learned in public schools. Perhaps the most important is one not often mentioned, and that is the ability to convert an important business question into an appropriate mathematical one, to solve the mathematical problem, and then to explain the implications of the solution for the original business problem. This ability to inhabit simultaneously the business world and the mathematical world, to translate between the two, and, as a consequence, to bring clarity to complex, real-world issues is of extraordinary importance.

In addition, the participants in this conversation understood and interpreted graphs and tables, computed, approximated, estimated, interpolated, extrapolated, used probabilistic concepts to draw conclusions, generalized from

small samples to large populations, identified the limits of their analyses, discovered relationships, recognized and used variables and functions, analyzed and compared data sets, and created and interpreted models. Another very important aspect of their work was that they identified additional questions, and they suggested ways to shed light on those questions through additional analysis.

There were two broad issues in this conversation that required mathematical perspectives. The first was to develop as rigorous and cost effective a data collection and analysis process as was practical. It involved perhaps 10 different analysts who attacked the problem from different viewpoints. The process also required integration of the mathematical learnings of all 10 analysts and translation of the results into business language that could be understood by non-mathematicians.

The second broad issue was to understand from the perspective of the decision-makers who were listening to the presentation which results were most reliable, which were subject to reinterpretation, which were actually judgments not supported by appropriate analysis, and which were hypotheses that truly required more research. In addition, these business people would likely identify synergies in the research that were not contemplated by the analysts. These synergies need to be analyzed to determine if—mathematically—they were real. The most obvious one was where the inventory analysts said that the customers don't like to use the phone to place orders. This is bad news for the sales analysts who are counting on phone data collection to correct errors caused by language problems. Of course, we need more information to know the magnitude—or even the existance—of the problem.

In brief, the analyses that preceded the dialogue might each be considered a mathematical task in the business world:

  • A cost analysis of store operations and catalogs was conducted using data from existing American and possibly other operations.
  • Customer preferences research was analyzed to determine preferences in quality and life-style. The data collection itself could not be carried out by a high school graduate without guidance, but 80% of the analysis could.
  • Cultural differences were recognized as a causes of analytical error. Careful analysis required judgment. In addition, sources of data were identified in the U.S., and comparable sources were found lacking in Japan. A search was conducted for other comparable retail experience, but none was found. On the other hand, sales data from car parts and computers were assessed for relevance.
  • Rates of change are important in understanding how Japanese and American stores differ. Sales per square foot, price increases,
  • asset costs, labor costs and so forth were compared to American standards to determine whether a store based in Japan would be a viable business.
  • "Nielsen" style ratings of 1000 families were used to collect data. Sample size and error estimates were mentioned. Key drivers of behavior (lifestyle, income, education) were mentioned, but this list may not be complete. What needs to be known about these families to predict their buying behavior? What does "lifestyle" include? How would we quantify some of these variables?
  • A hypothesis was presented that catalog size and product diversity drive higher sales. What do we need to know to assess the validity of this hypothesis? Another hypothesis was presented about the quality of the translation. What was the evidence for this hypothesis? Is this a mathematical question? Sales may also be proportional to the amount of time a potential customer retains the catalog. How could one ascertain this?
  • Despite the abundance of data, much uncertainty remains about what to expect from sales over the first two years. Analysis could be conducted with the data about the possible inventory consequences of choosing the wrong scenario.
  • One might wonder about the uncertainty in size scales. What is so difficult about identifying the colors that Japanese people prefer? Can these preferences be predicted? Will this increase the complexity of the inventory management task?
  • Can we predict how many people will not use phones? What do they use instead?

As seen through a mathematical lens, the business world can be a rich, complex, and essentially limitless source of fascinating questions.

R OL F ESSENDEN is Vice-President of Inventory Planning and Control at L. L. Bean, Inc. He is also Co-Principal Investigator and Vice-Chair of Maine's State Systemic Initiative and Chair of the Strategic Planning Committee. He has previously served on the Mathematical Science Education Board, and on the National Alliance for State Science and Mathematics Coalitions (NASSMC).

3— Integrating Vocational and Academic Education

THOMAS BAILEY

Columbia University

In high school education, preparation for work immediately after high school and preparation for post-secondary education have traditionally been viewed as incompatible. Work-bound high-school students end up in vocational education tracks, where courses usually emphasize specific skills with little attention to underlying theoretical and conceptual foundations. 1 College-bound students proceed through traditional academic discipline-based courses, where they learn English, history, science, mathematics, and foreign languages, with only weak and often contrived references to applications of these skills in the workplace or in the community outside the school. To be sure, many vocational teachers do teach underlying concepts, and many academic teachers motivate their lessons with examples and references to the world outside the classroom. But these enrichments are mostly frills, not central to either the content or pedagogy of secondary school education.

Rethinking Vocational and Academic Education

Educational thinking in the United States has traditionally placed priority on college preparation. Thus the distinct track of vocational education has been seen as an option for those students who are deemed not capable of success in the more desirable academic track. As vocational programs acquired a reputation

as a ''dumping ground," a strong background in vocational courses (especially if they reduced credits in the core academic courses) has been viewed as a threat to the college aspirations of secondary school students.

This notion was further reinforced by the very influential 1983 report entitled A Nation at Risk (National Commission on Excellence in Education, 1983), which excoriated the U.S. educational system for moving away from an emphasis on core academic subjects that, according to the report, had been the basis of a previously successful American education system. Vocational courses were seen as diverting high school students from core academic activities. Despite the dubious empirical foundation of the report's conclusions, subsequent reforms in most states increased the number of academic courses required for graduation and reduced opportunities for students to take vocational courses.

The distinction between vocational students and college-bound students has always had a conceptual flaw. The large majority of students who go to four-year colleges are motivated, at least to a significant extent, by vocational objectives. In 1994, almost 247,000 bachelors degrees were conferred in business administration. That was only 30,000 less than the total number (277,500) of 1994 bachelor degree conferred in English, mathematics, philosophy, religion, physical sciences and science technologies, biological and life sciences, social sciences, and history combined . Furthermore, these "academic" fields are also vocational since many students who graduate with these degrees intend to make their living working in those fields.

Several recent economic, technological, and educational trends challenge this sharp distinction between preparation for college and for immediate post-high-school work, or, more specifically, challenge the notion that students planning to work after high school have little need for academic skills while college-bound students are best served by an abstract education with only tenuous contact with the world of work:

  • First, many employers and analysts are arguing that, due to changes in the nature of work, traditional approaches to teaching vocational skills may not be effective in the future. Given the increasing pace of change and uncertainty in the workplace, young people will be better prepared, even for entry level positions and certainly for subsequent positions, if they have an underlying understanding of the scientific, mathematical, social, and even cultural aspects of the work that they will do. This has led to a growing emphasis on integrating academic and vocational education. 2
  • Views about teaching and pedagogy have increasingly moved toward a more open and collaborative "student-centered" or "constructivist" teaching style that puts a great deal of emphasis on having students work together on complex, open-ended projects. This reform strategy is now widely implemented through the efforts of organizations such as the Coalition of Essential Schools, the National Center for Restructuring Education, Schools, and Teaching at
  • Teachers College, and the Center for Education Research at the University of Wisconsin at Madison. Advocates of this approach have not had much interaction with vocational educators and have certainly not advocated any emphasis on directly preparing high school students for work. Nevertheless, the approach fits well with a reformed education that integrates vocational and academic skills through authentic applications. Such applications offer opportunities to explore and combine mathematical, scientific, historical, literary, sociological, economic, and cultural issues.
  • In a related trend, the federal School-to-Work Opportunities Act of 1994 defines an educational strategy that combines constructivist pedagogical reforms with guided experiences in the workplace or other non-work settings. At its best, school-to-work could further integrate academic and vocational learning through appropriately designed experiences at work.
  • The integration of vocational and academic education and the initiatives funded by the School-to-Work Opportunities Act were originally seen as strategies for preparing students for work after high school or community college. Some educators and policy makers are becoming convinced that these approaches can also be effective for teaching academic skills and preparing students for four-year college. Teaching academic skills in the context of realistic and complex applications from the workplace and community can provide motivational benefits and may impart a deeper understanding of the material by showing students how the academic skills are actually used. Retention may also be enhanced by giving students a chance to apply the knowledge that they often learn only in the abstract. 3
  • During the last twenty years, the real wages of high school graduates have fallen and the gap between the wages earned by high school and college graduates has grown significantly. Adults with no education beyond high school have very little chance of earning enough money to support a family with a moderate lifestyle. 4 Given these wage trends, it seems appropriate and just that every high school student at least be prepared for college, even if some choose to work immediately after high school.

Innovative Examples

There are many examples of programs that use work-related applications both to teach academic skills and to prepare students for college. One approach is to organize high school programs around broad industrial or occupational areas, such as health, agriculture, hospitality, manufacturing, transportation, or the arts. These broad areas offer many opportunities for wide-ranging curricula in all academic disciplines. They also offer opportunities for collaborative work among teachers from different disciplines. Specific skills can still be taught in this format but in such a way as to motivate broader academic and theoretical themes. Innovative programs can now be found in many vocational

high schools in large cities, such as Aviation High School in New York City and the High School of Agricultural Science and Technology in Chicago. Other schools have organized schools-within-schools based on broad industry areas.

Agriculturally based activities, such as 4H and Future Farmers of America, have for many years used the farm setting and students' interest in farming to teach a variety of skills. It takes only a little imagination to think of how to use the social, economic, and scientific bases of agriculture to motivate and illustrate skills and knowledge from all of the academic disciplines. Many schools are now using internships and projects based on local business activities as teaching tools. One example among many is the integrated program offered by the Thomas Jefferson High School for Science and Technology in Virginia, linking biology, English, and technology through an environmental issues forum. Students work as partners with resource managers at the Mason Neck National Wildlife Refuge and the Mason Neck State Park to collect data and monitor the daily activities of various species that inhabit the region. They search current literature to establish a hypothesis related to a real world problem, design an experiment to test their hypothesis, run the experiment, collect and analyze data, draw conclusions, and produce a written document that communicates the results of the experiment. The students are even responsible for determining what information and resources are needed and how to access them. Student projects have included making plans for public education programs dealing with environmental matters, finding solutions to problems caused by encroaching land development, and making suggestions for how to handle the overabundance of deer in the region.

These examples suggest the potential that a more integrated education could have for all students. Thus continuing to maintain a sharp distinction between vocational and academic instruction in high school does not serve the interests of many of those students headed for four-year or two-year college or of those who expect to work after high school. Work-bound students will be better prepared for work if they have stronger academic skills, and a high-quality curriculum that integrates school-based learning into work and community applications is an effective way to teach academic skills for many students.

Despite the many examples of innovative initiatives that suggest the potential for an integrated view, the legacy of the duality between vocational and academic education and the low status of work-related studies in high school continue to influence education and education reform. In general, programs that deviate from traditional college-prep organization and format are still viewed with suspicion by parents and teachers focused on four-year college. Indeed, college admissions practices still very much favor the traditional approaches. Interdisciplinary courses, "applied" courses, internships, and other types of work experience that characterize the school-to-work strategy or programs that integrate academic and vocational education often do not fit well into college admissions requirements.

Joining Work and Learning

What implications does this have for the mathematics standards developed by the National Council of Teachers of Mathematics (NCTM)? The general principle should be to try to design standards that challenge rather than reinforce the distinction between vocational and academic instruction. Academic teachers of mathematics and those working to set academic standards need to continue to try to understand the use of mathematics in the workplace and in everyday life. Such understandings would offer insights that could suggest reform of the traditional curriculum, but they would also provide a better foundation for teaching mathematics using realistic applications. The examples in this volume are particularly instructive because they suggest the importance of problem solving, logic, and imagination and show that these are all important parts of mathematical applications in realistic work settings. But these are only a beginning.

In order to develop this approach, it would be helpful if the NCTM standards writers worked closely with groups that are setting industry standards. 5 This would allow both groups to develop a deeper understanding of the mathematics content of work.

The NCTM's Curriculum Standards for Grades 9-12 include both core standards for all students and additional standards for "college-intending" students. The argument presented in this essay suggests that the NCTM should dispense with the distinction between college intending and non-college intending students. Most of the additional standards, those intended only for the "college intending" students, provide background that is necessary or beneficial for the calculus sequence. A re-evaluation of the role of calculus in the high school curriculum may be appropriate, but calculus should not serve as a wedge to separate college-bound from non-college-bound students. Clearly, some high school students will take calculus, although many college-bound students will not take calculus either in high school or in college. Thus in practice, calculus is not a characteristic that distinguishes between those who are or are not headed for college. Perhaps standards for a variety of options beyond the core might be offered. Mathematics standards should be set to encourage stronger skills for all students and to illustrate the power and usefulness of mathematics in many settings. They should not be used to institutionalize dubious distinctions between groups of students.

Bailey, T. & Merritt, D. (1997). School-to-work for the collegebound . Berkeley, CA: National Center for Research in Vocational Education.

Hoachlander, G . (1997) . Organizing mathematics education around work . In L.A. Steen (Ed.), Why numbers count: Quantitative literacy for tomorrow's America , (pp. 122-136). New York: College Entrance Examination Board.

Levy, F. & Murnane, R. (1992). U.S. earnings levels and earnings inequality: A review of recent trends and proposed explanations. Journal of Economic Literature , 30 , 1333-1381.

National Commission on Excellence in Education. (1983). A nation at risk: The imperative for educational reform . Washington, DC: Author.

T HOMAS B AILEY is an Associate Professor of Economics Education at Teachers College, Columbia University. He is also Director of the Institute on Education and the Economy and Director of the Community College Research Center, both at Teachers College. He is also on the board of the National Center for Research in Vocational Education.

4— The Importance of Workplace and Everyday Mathematics

JEAN E. TAYLOR

Rutgers University

For decades our industrial society has been based on fossil fuels. In today's knowledge-based society, mathematics is the energy that drives the system. In the words of the new WQED television series, Life by the Numbers , to create knowledge we "burn mathematics." Mathematics is more than a fixed tool applied in known ways. New mathematical techniques and analyses and even conceptual frameworks are continually required in economics, in finance, in materials science, in physics, in biology, in medicine.

Just as all scientific and health-service careers are mathematically based, so are many others. Interaction with computers has become a part of more and more jobs, and good analytical skills enhance computer use and troubleshooting. In addition, virtually all levels of management and many support positions in business and industry require some mathematical understanding, including an ability to read graphs and interpret other information presented visually, to use estimation effectively, and to apply mathematical reasoning.

What Should Students Learn for Today's World?

Education in mathematics and the ability to communicate its predictions is more important than ever for moving from low-paying jobs into better-paying ones. For example, my local paper, The Times of Trenton , had a section "Focus

on Careers" on October 5, 1997 in which the majority of the ads were for high technology careers (many more than for sales and marketing, for example).

But precisely what mathematics should students learn in school? Mathematicians and mathematics educators have been discussing this question for decades. This essay presents some thoughts about three areas of mathematics—estimation, trigonometry, and algebra—and then some thoughts about teaching and learning.

Estimation is one of the harder skills for students to learn, even if they experience relatively little difficulty with other aspects of mathematics. Many students think of mathematics as a set of precise rules yielding exact answers and are uncomfortable with the idea of imprecise answers, especially when the degree of precision in the estimate depends on the context and is not itself given by a rule. Yet it is very important to be able to get an approximate sense of the size an answer should be, as a way to get a rough check on the accuracy of a calculation (I've personally used it in stores to detect that I've been charged twice for the same item, as well as often in my own mathematical work), a feasibility estimate, or as an estimation for tips.

Trigonometry plays a significant role in the sciences and can help us understand phenomena in everyday life. Often introduced as a study of triangle measurement, trigonometry may be used for surveying and for determining heights of trees, but its utility extends vastly beyond these triangular applications. Students can experience the power of mathematics by using sine and cosine to model periodic phenomena such as going around and around a circle, going in and out with tides, monitoring temperature or smog components changing on a 24-hour cycle, or the cycling of predator-prey populations.

No educator argues the importance of algebra for students aiming for mathematically-based careers because of the foundation it provides for the more specialized education they will need later. Yet, algebra is also important for those students who do not currently aspire to mathematics-based careers, in part because a lack of algebraic skills puts an upper bound on the types of careers to which a student can aspire. Former civil rights leader Robert Moses makes a good case for every student learning algebra, as a means of empowering students and providing goals, skills, and opportunities. The same idea was applied to learning calculus in the movie Stand and Deliver . How, then, can we help all students learn algebra?

For me personally, the impetus to learn algebra was at least in part to learn methods of solution for puzzles. Suppose you have 39 jars on three shelves. There are twice as many jars on the second shelf as the first, and four more jars on the third shelf than on the second shelf. How many jars are there on each shelf? Such problems are not important by themselves, but if they show the students the power of an idea by enabling them to solve puzzles that they'd like to solve, then they have value. We can't expect such problems to interest all students. How then can we reach more students?

Workplace and Everyday Settings as a Way of Making Sense

One of the common tools in business and industry for investigating mathematical issues is the spreadsheet, which is closely related to algebra. Writing a rule to combine the elements of certain cells to produce the quantity that goes into another cell is doing algebra, although the variables names are cell names rather than x or y . Therefore, setting up spreadsheet analyses requires some of the thinking that algebra requires.

By exploring mathematics via tasks which come from workplace and everyday settings, and with the aid of common tools like spreadsheets, students are more likely to see the relevance of the mathematics and are more likely to learn it in ways that are personally meaningful than when it is presented abstractly and applied later only if time permits. Thus, this essay argues that workplace and everyday tasks should be used for teaching mathematics and, in particular, for teaching algebra. It would be a mistake, however, to rely exclusively on such tasks, just as it would be a mistake to teach only spreadsheets in place of algebra.

Communicating the results of an analysis is a fundamental part of any use of mathematics on a job. There is a growing emphasis in the workplace on group work and on the skills of communicating ideas to colleagues and clients. But communicating mathematical ideas is also a powerful tool for learning, for it requires the student to sharpen often fuzzy ideas.

Some of the tasks in this volume can provide the kinds of opportunities I am talking about. Another problem, with clear connections to the real world, is the following, taken from the book entitled Consider a Spherical Cow: A Course in Environmental Problem Solving , by John Harte (1988). The question posed is: How does biomagnification of a trace substance occur? For example, how do pesticides accumulate in the food chain, becoming concentrated in predators such as condors? Specifically, identify the critical ecological and chemical parameters determining bioconcentrations in a food chain, and in terms of these parameters, derive a formula for the concentration of a trace substance in each link of a food chain. This task can be undertaken at several different levels. The analysis in Harte's book is at a fairly high level, although it still involves only algebra as a mathematical tool. The task could be undertaken at a more simple level or, on the other hand, it could be elaborated upon as suggested by further exercises given in that book. And the students could then present the results of their analyses to each other as well as the teacher, in oral or written form.

Concepts or Procedures?

When teaching mathematics, it is easy to spend so much time and energy focusing on the procedures that the concepts receive little if any attention. When teaching algebra, students often learn the procedures for using the quadratic formula or for solving simultaneous equations without thinking of intersections of curves and lines and without being able to apply the procedures in unfamiliar settings. Even

when concentrating on word problems, students often learn the procedures for solving "coin problems" and "train problems" but don't see the larger algebraic context. The formulas and procedures are important, but are not enough.

When using workplace and everyday tasks for teaching mathematics, we must avoid falling into the same trap of focusing on the procedures at the expense of the concepts. Avoiding the trap is not easy, however, because just like many tasks in school algebra, mathematically based workplace tasks often have standard procedures that can be used without an understanding of the underlying mathematics. To change a procedure to accommodate a changing business climate, to respond to changes in the tax laws, or to apply or modify a procedure to accommodate a similar situation, however, requires an understanding of the mathematical ideas behind the procedures. In particular, a student should be able to modify the procedures for assessing energy usage for heating (as in Heating-Degree-Days, p. 54) in order to assess energy usage for cooling in the summer.

To prepare our students to make such modifications on their own, it is important to focus on the concepts as well as the procedures. Workplace and everyday tasks can provide opportunities for students to attach meaning to the mathematical calculations and procedures. If a student initially solves a problem without algebra, then the thinking that went into his or her solution can help him or her make sense out of algebraic approaches that are later presented by the teacher or by other students. Such an approach is especially appropriate for teaching algebra, because our teaching of algebra needs to reach more students (too often it is seen by students as meaningless symbol manipulation) and because algebraic thinking is increasingly important in the workplace.

An Example: The Student/Professor Problem

To illustrate the complexity of learning algebra meaningfully, consider the following problem from a study by Clement, Lockhead, & Monk (1981):

Write an equation for the following statement: "There are six times as many students as professors at this university." Use S for the number of students and P for the number of professors. (p. 288)

The authors found that of 47 nonscience majors taking college algebra, 57% got it wrong. What is more surprising, however, is that of 150 calculus-level students, 37% missed the problem.

A first reaction to the most common wrong answer, 6 S = P , is that the students simply translated the words of the problems into mathematical symbols without thinking more deeply about the situation or the variables. (The authors note that some textbooks instruct students to use such translation.)

By analyzing transcripts of interviews with students, the authors found this approach and another (faulty) approach, as well. These students often drew a diagram showing six students and one professor. (Note that we often instruct students to draw diagrams when solving word problems.) Reasoning

from the diagram, and regarding S and P as units, the student may write 6 S = P , just as we would correctly write 12 in. = 1 ft. Such reasoning is quite sensible, though it misses the fundamental intent in the problem statement that S is to represent the number of students, not a student.

Thus, two common suggestions for students—word-for-word translation and drawing a diagram—can lead to an incorrect answer to this apparently simple problem, if the students do not more deeply contemplate what the variables are intended to represent. The authors found that students who wrote and could explain the correct answer, S = 6 P , drew upon a richer understanding of what the equation and the variables represent.

Clearly, then, we must encourage students to contemplate the meanings of variables. Yet, part of the power and efficiency of algebra is precisely that one can manipulate symbols independently of what they mean and then draw meaning out of the conclusions to which the symbolic manipulations lead. Thus, stable, long-term learning of algebraic thinking requires both mastery of procedures and also deeper analytical thinking.

Paradoxically, the need for sharper analytical thinking occurs alongside a decreased need for routine arithmetic calculation. Calculators and computers make routine calculation easier to do quickly and accurately; cash registers used in fast food restaurants sometimes return change; checkout counters have bar code readers and payment takes place by credit cards or money-access cards.

So it is education in mathematical thinking, in applying mathematical computation, in assessing whether an answer is reasonable, and in communicating the results that is essential. Teaching mathematics via workplace and everyday problems is an approach that can make mathematics more meaningful for all students. It is important, however, to go beyond the specific details of a task in order to teach mathematical ideas. While this approach is particularly crucial for those students intending to pursue careers in the mathematical sciences, it will also lead to deeper mathematical understanding for all students.

Clement, J., Lockhead, J., & Monk, G. (1981). Translation difficulties in learning mathematics. American Mathematical Monthly , 88 , 286-290.

Harte, J. (1988). Consider a spherical cow: A course in environmental problem solving . York, PA: University Science Books.

J EAN E. T AYLOR is Professor of Mathematics at Rutgers, the State University of New Jersey. She is currently a member of the Board of Directors of the American Association for the Advancement of Science and formerly chaired its Section A Nominating Committee. She has served as Vice President and as a Member-at-Large of the Council of the American Mathematical Society, and served on its Executive Committee and its Nominating Committee. She has also been a member of the Joint Policy Board for Mathematics, and a member of the Board of Advisors to The Geometry Forum (now The Mathematics Forum) and to the WQED television series, Life by the Numbers .

5— Working with Algebra

DANIEL CHAZAN

Michigan State University

SANDRA CALLIS BETHELL

Holt High School

Teaching a mathematics class in which few of the students have demonstrated success is a difficult assignment. Many teachers avoid such assignments, when possible. On the one hand, high school mathematics teachers, like Bertrand Russell, might love mathematics and believe something like the following:

Mathematics, rightly viewed, possesses not only truth, but supreme beauty—a beauty cold and austere, like that of sculpture, without appeal to any part of our weaker nature, without the gorgeous trappings of painting or music, yet sublimely pure, and capable of a stern perfection such as only the greatest art can show. … Remote from human passions, remote even from the pitiful facts of nature, the generations have gradually created an ordered cosmos, where pure thought can dwell as in its nature home, and where one, at least, of our nobler impulses can escape from the dreary exile of the natural world. (Russell, 1910, p. 73)

But, on the other hand, students may not have the luxury, in their circumstances, of appreciating this beauty. Many of them may not see themselves as thinkers because contemplation would take them away from their primary

focus: how to get by in a world that was not created for them. Instead, like Jamaica Kincaid, they may be asking:

What makes the world turn against me and all who look like me? I won nothing, I survey nothing, when I ask this question, the luxury of an answer that will fill volumes does not stretch out before me. When I ask this question, my voice is filled with despair. (Kincaid, 1996, pp. 131-132)

Our Teaching and Issues it Raised

During the 1991-92 and 1992-93 school years, we (a high school teacher and a university teacher educator) team taught a lower track Algebra I class for 10th through 12th grade students. 1 Most of our students had failed mathematics before, and many needed to pass Algebra I in order to complete their high school mathematics requirement for graduation. For our students, mathematics had become a charged subject; it carried a heavy burden of negative experiences. Many of our students were convinced that neither they nor their peers could be successful in mathematics.

Few of our students did well in other academic subjects, and few were headed on to two- or four-year colleges. But the students differed in their affiliation with the high school. Some, called ''preppies" or "jocks" by others, were active participants in the school's activities. Others, "smokers" or "stoners," were rebelling to differing degrees against school and more broadly against society. There were strong tensions between members of these groups. 2

Teaching in this setting gives added importance and urgency to the typical questions of curriculum and motivation common to most algebra classes. In our teaching, we explored questions such as the following:

  • What is it that we really want high school students, especially those who are not college-intending, to study in algebra and why?
  • What is the role of algebra's manipulative skills in a world with graphing calculators and computers? How do the manipulative skills taught in the traditional curriculum give students a new perspective on, and insight into, our world?
  • If our teaching efforts depend on students' investment in learning, on what grounds can we appeal to them, implicitly or explicitly, for energy and effort? In a tracked, compulsory setting, how can we help students, with broad interests and talents and many of whom are not college-intending, see value in a shared exploration of algebra?

An Approach to School Algebra

As a result of thinking about these questions, in our teaching we wanted to avoid being in the position of exhorting students to appreciate the beauty or utility of algebra. Our students were frankly skeptical of arguments based on

utility. They saw few people in their community using algebra. We had also lost faith in the power of extrinsic rewards and punishments, like failing grades. Many of our students were skeptical of the power of the high school diploma to alter fundamentally their life circumstances. We wanted students to find the mathematical objects we were discussing in the world around them and thus learn to value the perspective that this mathematics might give them on their world.

To help us in this task, we found it useful to take what we call a "relationships between quantities" approach to school algebra. In this approach, the fundamental mathematical objects of study in school algebra are functions that can be represented by inputs and outputs listed in tables or sketched or plotted on graphs, as well as calculation procedures that can be written with algebraic symbols. 3 Stimulated, in part, by the following quote from August Comte, we viewed these functions as mathematical representations of theories people have developed for explaining relationships between quantities.

In the light of previous experience, we must acknowledge the impossibility of determining, by direct measurement, most of the heights and distances we should like to know. It is this general fact which makes the science of mathematics necessary. For in renouncing the hope, in almost every case, of measuring great heights or distances directly, the human mind has had to attempt to determine them indirectly, and it is thus that philosophers were led to invent mathematics. (Quoted in Serres, 1982, p. 85)

The "Sponsor" Project

Using this approach to the concept of function, during the 1992-93 school year, we designed a year-long project for our students. The project asked pairs of students to find the mathematical objects we were studying in the workplace of a community sponsor. Students visited the sponsor's workplace four times during the year—three after-school visits and one day-long excused absence from school. In these visits, the students came to know the workplace and learned about the sponsor's work. We then asked students to write a report describing the sponsor's workplace and answering questions about the nature of the mathematical activity embedded in the workplace. The questions are organized in Table 5-1 .

Using These Questions

In order to determine how the interviews could be structured and to provide students with a model, we chose to interview Sandra's husband, John Bethell, who is a coatings inspector for an engineering firm. When asked about his job, John responded, "I argue for a living." He went on to describe his daily work inspecting contractors painting water towers. Since most municipalities contract with the lowest bidder when a water tower needs to be painted, they will often hire an engineering firm to make sure that the contractor works according to specification. Since the contractor has made a low bid, there are strong

TABLE 5-1: Questions to ask in the workplace

financial incentives for the contractor to compromise on quality in order to make a profit.

In his work John does different kinds of inspections. For example, he has a magnetic instrument to check the thickness of the paint once it has been applied to the tower. When it gives a "thin" reading, contractors often question the technology. To argue for the reading, John uses the surface area of the tank, the number of paint cans used, the volume of paint in the can, and an understanding of the percentage of this volume that evaporates to calculate the average thickness of the dry coating. Other examples from his workplace involve the use of tables and measuring instruments of different kinds.

Some Examples of Students' Work

When school started, students began working on their projects. Although many of the sponsors initially indicated that there were no mathematical dimensions to their work, students often were able to show sponsors places where the mathematics we were studying was to be found. For example, Jackie worked with a crop and soil scientist. She was intrigued by the way in which measurement of weight is used to count seeds. First, her sponsor would weigh a test batch of 100 seeds to generate a benchmark weight. Then, instead of counting a large number of seeds, the scientist would weigh an amount of seeds and compute the number of seeds such a weight would contain.

Rebecca worked with a carpeting contractor who, in estimating costs, read the dimensions of rectangular rooms off an architect's blueprint, multiplied to find the area of the room in square feet (doing conversions where necessary), then multiplied by a cost per square foot (which depended on the type of carpet) to compute the cost of the carpet. The purpose of these estimates was to prepare a bid for the architect where the bid had to be as low as possible without making the job unprofitable. Rebecca used a chart ( Table 5-2 ) to explain this procedure to the class.

Joe and Mick, also working in construction, found out that in laying pipes, there is a "one by one" rule of thumb. When digging a trench for the placement of the pipe, the non-parallel sides of the trapezoidal cross section must have a slope of 1 foot down for every one foot across. This ratio guarantees that the dirt in the hole will not slide down on itself. Thus, if at the bottom of the hole, the trapezoid must have a certain width in order to fit the pipe, then on ground level the hole must be this width plus twice the depth of the hole. Knowing in advance how wide the hole must be avoids lengthy and costly trial and error.

Other students found that functions were often embedded in cultural artifacts found in the workplace. For example, a student who visited a doctor's office brought in an instrument for predicting the due dates of pregnant women, as well as providing information about average fetal weight and length ( Figure 5-1 ).

TABLE 5-2: Cost of carpet worksheet

math skills essay

FIGURE 5-1: Pregnancy wheel

While the complexities of organizing this sort of project should not be minimized—arranging sponsors, securing parental permission, and meeting administrators and parent concerns about the requirement of off-campus, after-school work—we remain intrigued by the potential of such projects for helping students see mathematics in the world around them. The notions of identifying central mathematical objects for a course and then developing ways of identifying those objects in students' experience seems like an important alternative to the use of application-based materials written by developers whose lives and social worlds may be quite different from those of students.

Chazen, D. (1996). Algebra for all students? Journal of Mathematical Behavior , 15 (4), 455-477.

Eckert, P. (1989). Jocks and burnouts: Social categories and identity in the high school . New York: Teachers College Press.

Fey, J. T., Heid, M. K., et al. (1995). Concepts in algebra: A technological approach . Dedham, MA: Janson Publications.

Kieran, C., Boileau, A., & Garancon, M. (1996). Introducing algebra by mean of a technology-supported, functional approach. In N. Bednarz et al. (Eds.), Approaches to algebra , (pp. 257-293). Kluwer Academic Publishers: Dordrecht, The Netherlands.

Kincaid, J. (1996). The autobiography of my mother . New York: Farrar, Straus, Giroux.

Nemirovsky, R. (1996). Mathematical narratives, modeling and algebra. In N. Bednarz et al. (Eds.) Approaches to algebra , (pp. 197-220). Kluwer Academic Publishers: Dordrecht, The Netherlands.

Russell, B. (1910). Philosophical Essays . London: Longmans, Green.

Schwartz, J. & Yerushalmy, M. (1992). Getting students to function in and with algebra. In G. Harel & E. Dubinsky (Eds.), The concept of function: Aspects of epistemology and pedagogy , (MAA Notes, Vol. 25, pp. 261-289). Washington, DC: Mathematical Association of America.

Serres, M. (1982). Mathematics and philosophy: What Thales saw … In J. Harari & D. Bell (Eds.), Hermes: Literature, science, philosophy , (pp. 84-97). Baltimore, MD: Johns Hopkins.

Thompson, P. (1993). Quantitative reasoning, complexity, and additive structures. Educational Studies in Mathematics , 25 , 165-208.

Yerushalmy, M. & Schwartz, J. L. (1993). Seizing the opportunity to make algebra mathematically and pedagogically interesting. In T. A. Romberg, E. Fennema, & T. P. Carpenter (Eds.), Integrating research on the graphical representation of functions , (pp. 41-68). Hillsdale, NJ: Lawrence Erlbaum Associates.

D ANIEL C HAZAN is an Associate Professor of Teacher Education at Michigan State University. To assist his research in mathematics teaching and learning, he has taught algebra at the high school level. His interests include teaching mathematics by examining student ideas, using computers to support student exploration, and the potential for the history and philosophy of mathematics to inform teaching.

S ANDRA C ALLIS B ETHELL has taught mathematics and Spanish at Holt High School for 10 years. She has also completed graduate work at Michigan State University and Western Michigan University. She has interest in mathematics reform, particularly in meeting the needs of diverse learners in algebra courses.

Emergency Calls

A city is served by two different ambulance companies. City logs record the date, the time of the call, the ambulance company, and the response time for each 911 call ( Table 1 ). Analyze these data and write a report to the City Council (with supporting charts and graphs) advising it on which ambulance company the 911 operators should choose to dispatch for calls from this region.

TABLE 1: Ambulance dispatch log sheet, May 1–30

This problem confronts the student with a realistic situation and a body of data regarding two ambulance companies' response times to emergency calls. The data the student is provided are typically "messy"—just a log of calls and response times, ordered chronologically. The question is how to make sense of them. Finding patterns in data such as these requires a productive mixture of mathematics common sense, and intellectual detective work. It's the kind of reasoning that students should be able to do—the kind of reasoning that will pay off in the real world.

Mathematical Analysis

In this case, a numerical analysis is not especially informative. On average, the companies are about the same: Arrow has a mean response time of 11.4 minutes compared to 11.6 minutes for Metro. The spread of the data is also not very helpful. The ranges of their distributions are exactly the same: from 6 minutes to 19 minutes. The standard deviation of Arrow's response time is a little longer—4.3 minutes versus 3.4 minutes for Metro—indicating that Arrow's response times fluctuate a bit more.

Graphs of the response times (Figures 1 and 2 ) reveal interesting features. Both companies, especially Arrow, seem to have bimodal distributions, which is to say that there are two clusters of data without much data in between.

math skills essay

FIGURE 1: Distribution of Arrow's response times

math skills essay

FIGURE 2: Distribution of Metro's response times

The distributions for both companies suggest that there are some other factors at work. Might a particular driver be the problem? Might the slow response times for either company be on particular days of the week or at particular times of day? Graphs of the response time versus the time of day (Figures 3 and 4 ) shed some light on these questions.

math skills essay

FIGURE 3: Arrow response times by time of day

math skills essay

FIGURE 4: Metro response times by time of day

These graphs show that Arrow's response times were fast except between 5:30 AM and 9:00 AM, when they were about 9 minutes slower on average. Similarly, Metro's response times were fast except between about 3:30 PM and 6:30 PM, when they were about 5 minutes slower. Perhaps the locations of the companies make Arrow more susceptible to the morning rush hour and Metro more susceptible to the afternoon rush hour. On the other hand, the employees on Arrow's morning shift or Metro's afternoon shift may not be efficient. To avoid slow responses, one could recommend to the City Council that Metro be called during the morning and that Arrow be called during the afternoon. A little detective work into the sources of the differences between the companies may yield a better recommendation.

Comparisons may be drawn between two samples in various contexts—response times for various services (taxis, computer-help desks, 24-hour hot lines at automobile manufacturers) being one class among many. Depending upon the circumstances, the data may tell very different stories. Even in the situation above, if the second pair of graphs hadn't offered such clear explanations, one might have argued that although the response times for Arrow were better on average the spread was larger, thus making their "extremes" more risky. The fundamental idea is using various analysis and representation techniques to make sense of data when the important factors are not necessarily known ahead of time.

Back-of-the-Envelope Estimates

Practice "back-of-the-envelope" estimates based on rough approximations that can be derived from common sense or everyday observations. Examples:

  • Consider a public high school mathematics teacher who feels that students should work five nights a week, averaging about 35 minutes a night, doing focused on-task work and who intends to grade all homework with comments and corrections. What is a reasonable number of hours per week that such a teacher should allocate for grading homework?
  • How much paper does The New York Times use in a week? A paper company that wishes to make a bid to become their sole supplier needs to know whether they have enough current capacity. If the company were to store a two-week supply of newspaper, will their empty 14,000 square foot warehouse be big enough?

Some 50 years ago, physicist Enrico Fermi asked his students at the University of Chicago, "How many piano tuners are there in Chicago?" By asking such questions, Fermi wanted his students to make estimates that involved rough approximations so that their goal would be not precision but the order of magnitude of their result. Thus, many people today call these kinds of questions "Fermi questions." These generally rough calculations often require little more than common sense, everyday observations, and a scrap of paper, such as the back of a used envelope.

Scientists and mathematicians use the idea of order of magnitude , usually expressed as the closest power of ten, to give a rough sense of the size of a quantity. In everyday conversation, people use a similar idea when they talk about "being in the right ballpark." For example, a full-time job at minimum wage yields an annual income on the order of magnitude of $10,000 or 10 4 dollars. Some corporate executives and professional athletes make annual salaries on the order of magnitude of $10,000,000 or 10 7 dollars. To say that these salaries differ by a factor of 1000 or 10 3 , one can say that they differ by three orders of magnitude. Such a lack of precision might seem unscientific or unmathematical, but such approximations are quite useful in determining whether a more precise measurement is feasible or necessary, what sort of action might be required, or whether the result of a calculation is "in the right ballpark." In choosing a strategy to protect an endangered species, for example, scientists plan differently if there are 500 animals remaining than if there are 5,000. On the other hand, determining whether 5,200 or 6,300 is a better estimate is not necessary, as the strategies will probably be the same.

Careful reasoning with everyday observations can usually produce Fermi estimates that are within an order of magnitude of the exact answer (if there is one). Fermi estimates encourage students to reason creatively with approximate quantities and uncertain information. Experiences with such a process can help an individual function in daily life to determine the reasonableness of numerical calculations, of situations or ideas in the workplace, or of a proposed tax cut. A quick estimate of some revenue- or profit-enhancing scheme may show that the idea is comparable to suggesting that General Motors enter the summer sidewalk lemonade market in your neighborhood. A quick estimate could encourage further investigation or provide the rationale to dismiss the idea.

Almost any numerical claim may be treated as a Fermi question when the problem solver does not have access to all necessary background information. In such a situation, one may make rough guesses about relevant numbers, do a few calculations, and then produce estimates.

The examples are solved separately below.

Grading Homework

Although many component factors vary greatly from teacher to teacher or even from week to week, rough calculations are not hard to make. Some important factors to consider for the teacher are: how many classes he or she teaches, how many students are in each of the classes, how much experience has the teacher had in general and has the teacher previously taught the classes, and certainly, as part of teaching style, the kind of homework the teacher assigns, not to mention the teacher's efficiency in grading.

Suppose the teacher has 5 classes averaging 25 students per class. Because the teacher plans to write corrections and comments, assume that the students' papers contain more than a list of answers—they show some student work and, perhaps, explain some of the solutions. Grading such papers might take as long as 10 minutes each, or perhaps even longer. Assuming that the teacher can grade them as quickly as 3 minutes each, on average, the teacher's grading time is:

math skills essay

This is an impressively large number, especially for a teacher who already spends almost 25 hours/week in class, some additional time in preparation, and some time meeting with individual students. Is it reasonable to expect teachers to put in that kind of time? What compromises or other changes might the teacher make to reduce the amount of time? The calculation above offers four possibilities: Reduce the time spent on each homework paper, reduce the number of students per class, reduce the number of classes taught each day, or reduce the number of days per week that homework will be collected. If the teacher decides to spend at most 2 hours grading each night, what is the total number of students for which the teacher should have responsibility? This calculation is a partial reverse of the one above:

math skills essay

If the teacher still has 5 classes, that would mean 8 students per class!

The New York Times

Answering this question requires two preliminary estimates: the circulation of The New York Times and the size of the newspaper. The answers will probably be different on Sundays. Though The New York Times is a national newspaper, the number of subscribers outside the New York metropolitan area is probably small compared to the number inside. The population of the New York metropolitan area is roughly ten million people. Since most families buy at most one copy, and not all families buy The New York Times , the circulation might be about 1 million newspapers each day. (A circulation of 500,000 seems too small and 2 million seems too big.) The Sunday and weekday editions probably have different

circulations, but assume that they are the same since they probably differ by less than a factor of two—much less than an order of magnitude. When folded, a weekday edition of the paper measures about 1/2 inch thick, a little more than 1 foot long, and about 1 foot wide. A Sunday edition of the paper is the same width and length, but perhaps 2 inches thick. For a week, then, the papers would stack 6 × 1/2 + 2 = 5 inches thick, for a total volume of about 1 ft × 1 ft × 5/12 ft = 0.5 ft 3 .

The whole circulation, then, would require about 1/2 million cubic feet of paper per week, or about 1 million cubic feet for a two-week supply.

Is the company's warehouse big enough? The paper will come on rolls, but to make the estimates easy, assume it is stacked. If it were stacked 10 feet high, the supply would require 100,000 square feet of floor space. The company's 14,000 square foot storage facility will probably not be big enough as its size differs by almost an order of magnitude from the estimate. The circulation estimate and the size of the newspaper estimate should each be within a factor of 2, implying that the 100,000 square foot estimate is off by at most a factor of 4—less than an order of magnitude.

How big a warehouse is needed? An acre is 43,560 square feet so about two acres of land is needed. Alternatively, a warehouse measuring 300 ft × 300 ft (the length of a football field in both directions) would contain 90,000 square feet of floor space, giving a rough idea of the size.

After gaining some experience with these types of problems, students can be encouraged to pay close attention to the units and to be ready to make and support claims about the accuracy of their estimates. Paying attention to units and including units as algebraic quantities in calculations is a common technique in engineering and the sciences. Reasoning about a formula by paying attention only to the units is called dimensional analysis.

Sometimes, rather than a single estimate, it is helpful to make estimates of upper and lower bounds. Such an approach reinforces the idea that an exact answer is not the goal. In many situations, students could first estimate upper and lower bounds, and then collect some real data to determine whether the answer lies between those bounds. In the traditional game of guessing the number of jelly beans in a jar, for example, all students should be able to estimate within an order of magnitude, or perhaps within a factor of two. Making the closest guess, however, involves some chance.

Fermi questions are useful outside the workplace. Some Fermi questions have political ramifications:

  • How many miles of streets are in your city or town? The police chief is considering increasing police presence so that every street is patrolled by car at least once every 4 hours.
  • When will your town fill up its landfill? Is this a very urgent matter for the town's waste management personnel to assess in depth?
  • In his 1997 State of the Union address, President Clinton renewed his call for a tax deduction of up to $10,000 for the cost of college tuition. He estimates that 16.5 million students stand to benefit. Is this a reasonable estimate of the number who might take advantage of the tax deduction? How much will the deduction cost in lost federal revenue?

Creating Fermi problems is easy. Simply ask quantitative questions for which there is no practical way to determine exact values. Students could be encouraged to make up their own. Examples are: ''How many oak trees are there in Illinois?" or "How many people in the U.S. ate chicken for dinner last night?" "If all the people in the world were to jump in the ocean, how much would it raise the water level?" Give students the opportunity to develop their own Fermi problems and to share them with each other. It can stimulate some real mathematical thinking.

Scheduling Elevators

In some buildings, all of the elevators can travel to all of the floors, while in others the elevators are restricted to stopping only on certain floors. What is the advantage of having elevators that travel only to certain floors? When is this worth instituting?

Scheduling elevators is a common example of an optimization problem that has applications in all aspects of business and industry. Optimal scheduling in general not only can save time and money, but it can contribute to safety (e.g., in the airline industry). The elevator problem further illustrates an important feature of many economic and political arguments—the dilemma of trying simultaneously to optimize several different needs.

Politicians often promise policies that will be the least expensive, save the most lives, and be best for the environment. Think of flood control or occupational safety rules, for example. When we are lucky, we can perhaps find a strategy of least cost, a strategy that saves the most lives, or a strategy that damages the environment least. But these might not be the same strategies: generally one cannot simultaneously satisfy two or more independent optimization conditions. This is an important message for students to learn, in order to become better educated and more critical consumers and citizens.

In the elevator problem, customer satisfaction can be emphasized by minimizing the average elevator time (waiting plus riding) for employees in an office building. Minimizing wait-time during rush hours means delivering many people quickly, which might be accomplished by filling the elevators and making few stops. During off-peak hours, however, minimizing wait-time means maximizing the availability of the elevators. There is no reason to believe that these two goals will yield the same strategy. Finding the best strategy for each is a mathematical problem; choosing one of the two strategies or a compromise strategy is a management decision, not a mathematical deduction.

This example serves to introduce a complex topic whose analysis is well within the range of high school students. Though the calculations require little more than arithmetic, the task puts a premium on the creation of reasonable alternative strategies. Students should recognize that some configurations (e.g., all but one elevator going to the top floor and the one going to all the others) do not merit consideration, while others are plausible. A systematic evaluation of all possible configurations is usually required to find the optimal solution. Such a systematic search of the possible solution space is important in many modeling situations where a formal optimal strategy is not known. Creating and evaluating reasonable strategies for the elevators is quite appropriate for high school student mathematics and lends itself well to thoughtful group effort. How do you invent new strategies? How do you know that you have considered all plausible strategies? These are mathematical questions, and they are especially amenable to group discussion.

Students should be able to use the techniques first developed in solving a simple case with only a few stories and a few elevators to address more realistic situations (e.g., 50 stories, five elevators). Using the results of a similar but simpler problem to model a more complicated problem is an important way to reason in mathematics. Students

need to determine what data and variables are relevant. Start by establishing the kind of building—a hotel, an office building, an apartment building? How many people are on the different floors? What are their normal destinations (e.g., primarily the ground floor or, perhaps, a roof-top restaurant). What happens during rush hours?

To be successful at the elevator task, students must first develop a mathematical model of the problem. The model might be a graphical representation for each elevator, with time on the horizontal axis and the floors represented on the vertical axis, or a tabular representation indicating the time spent on each floor. Students must identify the pertinent variables and make simplifying assumptions about which of the possible floors an elevator will visit.

This section works through some of the details in a particularly simple case. Consider an office building with six occupied floors, employing 240 people, and a ground floor that is not used for business. Suppose there are three elevators, each of which can hold 10 people. Further suppose that each elevator takes approximately 25 seconds to fill on the ground floor, then takes 5 seconds to move between floors and 15 seconds to open and close at each floor on which it stops.

Scenario One

What happens in the morning when everyone arrives for work? Assume that everyone arrives at approximately the same time and enters the elevators on the ground floor. If all elevators go to all floors and if the 240 people are evenly divided among all three elevators, each elevator will have to make 8 trips of 10 people each.

When considering a single trip of one elevator, assume for simplicity that 10 people get on the elevator at the ground floor and that it stops at each floor on the way up, because there may be an occupant heading to each floor. Adding 5 seconds to move to each floor and 15 seconds to stop yields 20 seconds for each of the six floors. On the way down, since no one is being picked up or let off, the elevator does not stop, taking 5 seconds for each of six floors for a total of 30 seconds. This round-trip is represented in Table 1 .

TABLE 1: Elevator round-trip time, Scenario one

Since each elevator makes 8 trips, the total time will be 1,400 seconds or 23 minutes, 20 seconds.

Scenario Two

Now suppose that one elevator serves floors 1–3 and, because of the longer trip, two elevators are assigned to floors 4–6. The elevators serving the top

TABLE 2: Elevator round-trip times, Scenario two

floors will save 15 seconds for each of floors 1–3 by not stopping. The elevator serving the bottom floors will save 20 seconds for each of the top floors and will save time on the return trip as well. The times for these trips are shown in Table 2 .

Assuming the employees are evenly distributed among the floors (40 people per floor), elevator A will transport 120 people, requiring 12 trips, and elevators B and C will transport 120 people, requiring 6 trips each. These trips will take 1200 seconds (20 minutes) for elevator A and 780 seconds (13 minutes) for elevators B and C, resulting in a small time savings (about 3 minutes) over the first scenario. Because elevators B and C are finished so much sooner than elevator A, there is likely a more efficient solution.

Scenario Three

The two round-trip times in Table 2 do not differ by much because the elevators move quickly between floors but stop at floors relatively slowly. This observation suggests that a more efficient arrangement might be to assign each elevator to a pair of floors. The times for such a scenario are listed in Table 3 .

Again assuming 40 employees per floor, each elevator will deliver 80 people, requiring 8 trips, taking at most a total of 920 seconds. Thus this assignment of elevators results in a time savings of almost 35% when compared with the 1400 seconds it would take to deliver all employees via unassigned elevators.

TABLE 3: Elevator round-trip times, Scenario three

Perhaps this is the optimal solution. If so, then the above analysis of this simple case suggests two hypotheses:

  • The optimal solution assigns each floor to a single elevator.
  • If the time for stopping is sufficiently larger than the time for moving between floors, each elevator should serve the same number of floors.

Mathematically, one could try to show that this solution is optimal by trying all possible elevator assignments or by carefully reasoning, perhaps by showing that the above hypotheses are correct. Practically, however, it doesn't matter because this solution considers only the morning rush hour and ignores periods of low use.

The assignment is clearly not optimal during periods of low use, and much of the inefficiency is related to the first hypothesis for rush hour optimization: that each floor is served by a single elevator. With this condition, if an employee on floor 6 arrives at the ground floor just after elevator C has departed, for example, she or he will have to wait nearly two minutes for elevator C to return, even if elevators A and B are idle. There are other inefficiencies that are not considered by focusing on the rush hour. Because each floor is served by a single elevator, an employee who wishes to travel from floor 3 to floor 6, for example, must go via the ground floor and switch elevators. Most employees would prefer more flexibility than a single elevator serving each floor.

At times when the elevators are not all busy, unassigned elevators will provide the quickest response and the greatest flexibility.

Because this optimal solution conflicts with the optimal rush hour solution, some compromise is necessary. In this simple case, perhaps elevator A could serve all floors, elevator B could serve floors 1-3, and elevator C could serve floors 4-6.

The second hypothesis, above, deserves some further thought. The efficiency of the rush hour solution Table 3 is due in part to the even division of employees among the floors. If employees were unevenly distributed with, say, 120 of the 240 people working on the top two floors, then elevator C would need to make 12 trips, taking a total of 1380 seconds, resulting in almost no benefit over unassigned elevators. Thus, an efficient solution in an actual building must take into account the distribution of the employees among the floors.

Because the stopping time on each floor is three times as large as the traveling time between floors (15 seconds versus 5 seconds), this solution effectively ignores the traveling time by assigning the same number of employees to each elevator. For taller buildings, the traveling time will become more significant. In those cases fewer employees should be assigned to the elevators that serve the upper floors than are assigned to the elevators that serve the lower floors.

The problem can be made more challenging by altering the number of elevators, the number of floors, and the number of individuals working on each floor. The rate of movement of elevators can be determined by observing buildings in the local area. Some elevators move more quickly than others. Entrance and exit times could also be measured by students collecting

data on local elevators. In a similar manner, the number of workers, elevators, and floors could be taken from local contexts.

A related question is, where should the elevators go when not in use? Is it best for them to return to the ground floor? Should they remain where they were last sent? Should they distribute themselves evenly among the floors? Or should they go to floors of anticipated heavy traffic? The answers will depend on the nature of the building and the time of day. Without analysis, it will not be at all clear which strategy is best under specific conditions. In some buildings, the elevators are controlled by computer programs that "learn" and then anticipate the traffic patterns in the building.

A different example that students can easily explore in detail is the problem of situating a fire station or an emergency room in a city. Here the key issue concerns travel times to the region being served, with conflicting optimization goals: average time vs. maximum time. A location that minimizes the maximum time of response may not produce the least average time of response. Commuters often face similar choices in selecting routes to work. They may want to minimize the average time, the maximum time, or perhaps the variance, so that their departure and arrival times are more predictable.

Most of the optimization conditions discussed so far have been expressed in units of time. Sometimes, however, two optimization conditions yield strategies whose outcomes are expressed in different (and sometimes incompatible) units of measurement. In many public policy issues (e.g., health insurance) the units are lives and money. For environmental issues, sometimes the units themselves are difficult to identify (e.g., quality of life).

When one of the units is money, it is easy to find expensive strategies but impossible to find ones that have virtually no cost. In some situations, such as airline safety, which balances lives versus dollars, there is no strategy that minimize lives lost (since additional dollars always produce slight increases in safety), and the strategy that minimizes dollars will be at $0. Clearly some compromise is necessary. Working with models of different solutions can help students understand the consequences of some of the compromises.

Heating-Degree-Days

An energy consulting firm that recommends and installs insulation and similar energy saving devices has received a complaint from a customer. Last summer she paid $540 to insulate her attic on the prediction that it would save 10% on her natural gas bills. Her gas bills have been higher than the previous winter, however, and now she wants a refund on the cost of the insulation. She admits that this winter has been colder than the last, but she had expected still to see some savings.

The facts: This winter the customer has used 1,102 therms, whereas last winter she used only 1,054 therms. This winter has been colder: 5,101 heating-degree-days this winter compared to 4,201 heating-degree-days last winter. (See explanation below.) How does a representative of the energy consulting firm explain to this customer that the accumulated heating-degree-days measure how much colder this winter has been, and then explain how to calculate her anticipated versus her actual savings.

Explaining the mathematics behind a situation can be challenging and requires a real knowledge of the context, the procedures, and the underlying mathematical concepts. Such communication of mathematical ideas is a powerful learning device for students of mathematics as well as an important skill for the workplace. Though the procedure for this problem involves only proportions, a thorough explanation of the mathematics behind the procedure requires understanding of linear modeling and related algebraic reasoning, accumulation and other precursors of calculus, as well as an understanding of energy usage in home heating.

The customer seems to understand that a straight comparison of gas usage does not take into account the added costs of colder weather, which can be significant. But before calculating any anticipated or actual savings, the customer needs some understanding of heating-degree-days. For many years, weather services and oil and gas companies have been using heating-degree-days to explain and predict energy usage and to measure energy savings of insulation and other devices. Similar degree-day units are also used in studying insect populations and crop growth. The concept provides a simple measure of the accumulated amount of cold or warm weather over time. In the discussion that follows, all temperatures are given in degrees Fahrenheit, although the process is equally workable using degrees Celsius.

Suppose, for example, that the minimum temperature in a city on a given day is 52 degrees and the maximum temperature is 64 degrees. The average temperature for the day is then taken to be 58 degrees. Subtracting that result from 65 degrees (the cutoff point for heating), yields 7 heating-degree-days for the day. By recording high and low temperatures and computing their average each day, heating-degree-days can be accumulated over the course of a month, a winter, or any period of time as a measure of the coldness of that period.

Over five consecutive days, for example, if the average temperatures were 58, 50, 60, 67, and 56 degrees Fahrenheit, the calculation yields 7, 15, 5, 0, and 9 heating-degree-days respectively, for a total accumulation of 36 heating-degree-days for the five days. Note that the fourth day contributes 0 heating-degree-days to the total because the temperature was above 65 degrees.

The relationship between average temperatures and heating-degree-days is represented graphically in Figure 1 . The average temperatures are shown along the solid line graph. The area of each shaded rectangle represents the number of heating-degree-days for that day, because the width of each rectangle is one day and the height of each rectangle is the number of degrees below 65 degrees. Over time, the sum of the areas of the rectangles represents the number of heating-degree-days accumulated during the period. (Teachers of calculus will recognize connections between these ideas and integral calculus.)

The statement that accumulated heating-degree-days should be proportional to gas or heating oil usage is based primarily on two assumptions: first, on a day for which the average temperature is above 65 degrees, no heating should be required, and therefore there should be no gas or heating oil usage; second, a day for which the average temperature is 25 degrees (40 heating-degree-days) should require twice as much heating as a day for which the average temperature is 45

math skills essay

FIGURE 1: Daily heating-degree-days

degrees (20 heating-degree-days) because there is twice the temperature difference from the 65 degree cutoff.

The first assumption is reasonable because most people would not turn on their heat if the temperature outside is above 65 degrees. The second assumption is consistent with Newton's law of cooling, which states that the rate at which an object cools is proportional to the difference in temperature between the object and its environment. That is, a house which is 40 degrees warmer than its environment will cool at twice the rate (and therefore consume energy at twice the rate to keep warm) of a house which is 20 degrees warmer than its environment.

The customer who accepts the heating-degree-day model as a measure of energy usage can compare this winter's usage with that of last winter. Because 5,101/4,201 = 1.21, this winter has been 21% colder than last winter, and therefore each house should require 21% more heat than last winter. If this customer hadn't installed the insulation, she would have required 21% more heat than last year, or about 1,275 therms. Instead, she has required only 5% more heat (1,102/1,054 = 1.05), yielding a savings of 14% off what would have been required (1,102/1,275 = .86).

Another approach to this would be to note that last year the customer used 1,054 therms/4,201 heating-degree-days = .251 therms/heating-degree-day, whereas this year she has used 1,102 therms/5,101 heating-degree-days = .216 therms/heating-degree-day, a savings of 14%, as before.

How good is the heating-degree-day model in predicting energy usage? In a home that has a thermometer and a gas meter or a gauge on a tank, students could record daily data for gas usage and high and low temperature to test the accuracy of the model. Data collection would require only a few minutes per day for students using an electronic indoor/outdoor thermometer that tracks high and low temperatures. Of course, gas used for cooking and heating water needs to be taken into account. For homes in which the gas tank has no gauge or doesn't provide accurate enough data, a similar experiment could be performed relating accumulated heating-degree-days to gas or oil usage between fill-ups.

It turns out that in well-sealed modern houses, the cutoff temperature for heating can be lower than 65 degrees (sometimes as low as 55 degrees) because of heat generated by light bulbs, appliances, cooking, people, and pets. At temperatures sufficiently below the cutoff, linearity turns out to be a good assumption. Linear regression on the daily usage data (collected as suggested above) ought to find an equation something like U = -.251( T - 65), where T is the average temperature and U is the gas usage. Note that the slope, -.251, is the gas usage per heating-degree-day, and 65 is the cutoff. Note also that the accumulation of heating-degree-days takes a linear equation and turns it into a proportion. There are some important data analysis issues that could be addressed by such an investigation. It is sometimes dangerous, for example, to assume linearity with only a few data points, yet this widely used model essentially assumes linearity from only one data point, the other point having coordinates of 65 degrees, 0 gas usage.

Over what range of temperatures, if any, is this a reasonable assumption? Is the standard method of computing average temperature a good method? If, for example, a day is mostly near 20 degrees but warms up to 50 degrees for a short time in the afternoon, is 35 heating-degree-days a good measure of the heating required that day? Computing averages of functions over time is a standard problem that can be solved with integral calculus. With knowledge of typical and extreme rates of temperature change, this could become a calculus problem or a problem for approximate solution by graphical methods without calculus, providing background experience for some of the important ideas in calculus.

Students could also investigate actual savings after insulating a home in their school district. A customer might typically see 8-10% savings for insulating roofs, although if the house is framed so that the walls act like chimneys, ducting air from the house and the basement into the attic, there might be very little savings. Eliminating significant leaks, on the other hand, can yield savings of as much as 25%.

Some U.S. Department of Energy studies discuss the relationship between heating-degree-days and performance and find the cutoff temperature to be lower in some modern houses. State energy offices also have useful documents.

What is the relationship between heating-degree-days computed using degrees Fahrenheit, as above, and heating-degree-days computed using degrees Celsius? Showing that the proper conversion is a direct proportion and not the standard Fahrenheit-Celsius conversion formula requires some careful and sophisticated mathematical thinking.

Traditionally, vocational mathematics and precollege mathematics have been separate in schools. But the technological world in which today's students will work and live calls for increasing connection between mathematics and its applications. Workplace-based mathematics may be good mathematics for everyone.

High School Mathematics at Work illuminates the interplay between technical and academic mathematics. This collection of thought-provoking essays—by mathematicians, educators, and other experts—is enhanced with illustrative tasks from workplace and everyday contexts that suggest ways to strengthen high school mathematical education.

This important book addresses how to make mathematical education of all students meaningful—how to meet the practical needs of students entering the work force after high school as well as the needs of students going on to postsecondary education.

The short readable essays frame basic issues, provide background, and suggest alternatives to the traditional separation between technical and academic mathematics. They are accompanied by intriguing multipart problems that illustrate how deep mathematics functions in everyday settings—from analysis of ambulance response times to energy utilization, from buying a used car to "rounding off" to simplify problems.

The book addresses the role of standards in mathematics education, discussing issues such as finding common ground between science and mathematics education standards, improving the articulation from school to work, and comparing SAT results across settings.

Experts discuss how to develop curricula so that students learn to solve problems they are likely to encounter in life—while also providing them with approaches to unfamiliar problems. The book also addresses how teachers can help prepare students for postsecondary education.

For teacher education the book explores the changing nature of pedagogy and new approaches to teacher development. What kind of teaching will allow mathematics to be a guide rather than a gatekeeper to many career paths? Essays discuss pedagogical implication in problem-centered teaching, the role of complex mathematical tasks in teacher education, and the idea of making open-ended tasks—and the student work they elicit—central to professional discourse.

High School Mathematics at Work presents thoughtful views from experts. It identifies rich possibilities for teaching mathematics and preparing students for the technological challenges of the future. This book will inform and inspire teachers, teacher educators, curriculum developers, and others involved in improving mathematics education and the capabilities of tomorrow's work force.

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Why early math is just as important as early reading

by: Hank Pellissier | Updated: February 27, 2023

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Why early math is just as important as early reading

According to new research, the importance of mathematics in early childhood cannot be underestimated. So what grade do U.S. preschools deserve in math instruction?

Answer: F (At least most of them.)

Math is nearly absent in American preschools and prekindergarten classes. One study calculated that at preschools where kids spend six hours a day, math gets an average of only 58 seconds per day . Not even a full minute.

What’s more, those spare moments of math are often taught incorrectly . Learning to recite numbers from one to 10 doesn’t get kids very far, because often kids are just memorizing , according to Stanford math professor Jo Boaler, which does little to lay the groundwork for future problem solving and logical thought.

From “Talk, Read, Sing!” campaigns to closing the 30 million word gap, American parents, educators, and policymakers have embraced the importance of early literacy, yet we collectively presume it’s fine to tackle math later. Meanwhile, research clearly shows that early math exposure is crucial for later success in math.

Why is mathematics important in early childhood? “Early math skills have the greatest predictive power, followed by reading and then attention skills,” reports a psychology squad led by Greg J. Duncan , in School readiness and later achievement , published in Developmental Psychology in 2007. Follow-up studies continue to confirm the importance of early math skills. The more math-oriented activities kids do before kindergarten, the better they’ll understand math in school. Early math skills foretell higher aptitude in high school math and higher rates of college enrollment. And a 2014 Vanderbilt study determined that for “both males and females, mathematical precocity early in life predicts later creative contributions and leadership in critical occupational roles.”

Why counting doesn’t cut it

To date, the most common approach to teaching early math skills has been to surround young kids with numbers alongside their letters and encourage them to practice counting just as they practice singing the alphabet. But researchers say that this approach is short-changing children. In her essay “ Math Matters, Even for Little Kids ,” Stanford professor Deborah Stipek (co-written with Alan Schoenfeld and Deanna Gomby) explains the parallel to the alphabet: “Learning to count by rote teaches children number words and order, but it does not teach them number sense, any more than singing the letters L-M-N-O-P in the alphabet song teaches phonemic awareness.”

As for the magic of counting to 10, University of Chicago professor Susan Levine explains: “Kids can rattle off their numbers early, often from 1 to 10, and parents are surprised and impressed. But it’s a list with no meaning. When you say ‘give me 3 fish,’ they give you a handful.”

While parents and preschool teachers reinforce literacy lessons daily by reading together, singing, pointing out letters and letter sounds, math exposure often begins and ends with the counting.

Early math skills kids are born with

Evidence points to early addition and subtraction being an innate ability . In a 1992 study at the University of Arizona , for example, 6-month-old babies were shown one baby doll. As the babies watched, a screen was placed in front of the doll and then a second doll was placed behind the screen. When the screen was removed, scientists could tell that, at just 6 months old, babies expected to see two dolls. In instances when there were fewer or more dolls when the screen was removed, the babies stared longer because the results were wrong, a “violation of expectation.”

“ Subitize ” — from the Latin word for “suddenly” — is the ability to quickly identify the number of items in a small group. When Dustin Hoffman’s character in Rainman looked down at the pile of spilled toothpicks and knew without counting that there were 246, that was an example of advanced subitization. Preschoolers can differentiate between one and three items; by age 7, this increases to between four and seven items. It’s more than just a cool party trick: Research indicates that developing the ability to subitize larger numbers can increase math skills in a few ways . One example is “counting on” or being able to start at 5 and continue counting up, which is a math strategy first graders will need as they begin tackling adding and subtracting.

Numbers in new and different contexts

One way to build on kids’ innate math abilities is to focus on helping them count in contexts that are meaningful to them. To practice counting on, start with a number they recognize, like two toy dinosaurs. Add another and say, “three,” then add another and say “four,” helping them to connect the number names with the increase in objects.

Toddlers are natural sorters. By age 2, they start recognizing and making comparisons, such as more, same, and different. They enjoy organizing multiple objects into specific “sets”, i.e., groups or categories. Asking a child to sort their prized dinosaurs into groups, such as big and small, is the basis of a few important early math skills. First, they can compare the groups. (There are more small dinosaurs.) They can count how many are in each group (two big ones, six small ones). They can re-sort in different ways (the ones with spikes here, the ones without spikes there; arranging the dinosaurs in order from shortest to tallest). Numeracy is an intimidating word, but it simply means understanding what each number represents and beginning to understand the implications of number operations: What happens when you take one of the dinosaurs away?

The National Association for Education of Young Children notes that young children are building scientific inquiry skills when they sort, compare, describe, and put things in order in terms of observable characteristics, like the dinosaur’s height.

Amazing measuring

Children are rightfully fascinated by variability in size. They delight in the enormity of elephants, trees, and skyscrapers, and the minuteness of ants and caterpillars. The allure of discovering what’s comparatively bigger or smaller fosters their curiosity about inches, pounds, gallons, miles, and other systems of measurement. And that’s what parents and educators should encourage, according to Douglas H. Clements and Julie Sarama, authors of Learning and Teaching Early Math: The Learning Trajectories Approach . Clements and Sarama suggest that measurement is the best way for young kids to learn about math. They go so far as to say it’s better than counting. “We use length consistently in our everyday lives,” they write. “[Measurement] can help develop other areas… including reasoning and logic. Also, by its very nature, [measuring] connects two critical domains of early mathematics: geometry and number.”

Building blocks and the language of space

The next time you’re cleaning up your child’s blocks or Legos, just remember: they’re building their math brains. Boosting spatial skills via block play has been proven beneficial in many studies. For example, the complexity of a child’s LEGO play during the preschool years is correlated with higher math achievement in high school .

Exposing preschoolers to geometrical shapes including circles, squares, triangles, and rectangles helps them build a skill called visual literacy. Researchers Clements and Sarama discovered in one study that kids who learned shapes and spatial skills also showed pronounced benefits in math and writing readiness and even increased their IQ scores. (Related: How to teach your preschooler shapes and spatial skills .)

Clements urges parents and teachers to teach kids what he calls the “Language of space” – words like front, back, behind, top, bottom, over, under, last, first, next, backward, in, on, deep, shallow, triangle, square, corner, edge, etc. Stipek, former dean at the Stanford Graduate School of Education , suggests “when you’re reading a picture book to your child, point out position and spatial representation. Say, ‘the tree is behind the car’ and ‘the roof is a triangle.’” Helping a preschooler understand these relative terms is more than a math vocabulary lesson, says parent educator Nancy Gnass. It’s hard for a child to follow spatial directions commonly given to kindergartners – stand behind the blue line, put the blocks on bottom shelf, this is a quiet corner – if they don’t know what behind, bottom, or corner mean, she says. Preparing your child for directional language they’ll be expected to comply with regularly once they start school can head off misunderstandings and perceived behavior issues, like not understanding rather than deliberately not following directions, she says.

Patterns aren’t just pretty

Visual art and dance provide excellent ways to teach patterns, defined in A Math Dictionary for Kids  as ”a repeated design or recurring sequence.” According to Zero to Three, recognizing and creating patterns helps children learn to make predictions, understand what comes next, make logical connections, and use reasoning skills. Kids start to put together the “growing pattern” in counting and “relationship pattern” that’s the basis for multiplication.

Movement patterns can also imbue a trip to the park with mathematical benefits. Encourage your kinetic kid to walk-tiptoe-jump-repeat or skip-hop-run-in-a-circle-repeat, or stop, drop, and roll; repeat until they’re exhausted (and educated). (Related: Cool ways to teach your preschooler patterns )

Bring on the math games

The best way for parents to “mathematize” their children is to use math in the routines of daily life, either as games or as entertaining ways to solve problems. “Make math fun!” advises Eric Wilson, Lead Teacher at Pacific Primary School in San Francisco . “Young children work very hard when they’re playing. Play is the perfect learning environment.” Puzzles , building blocks , board games, and card games have all been studied, with researchers concluding that all of these elevate math skills.

Chutes and Ladders, for example, excels at teaching numbers, says Stanford professor Deborah J. Stipek, and playing with dice teaches addition.

Attitude matters

Possibly the most important thing any parent or teacher can do for preschoolers’ early math skills is encourage children to believe that they can succeed . “Self-efficacy, which is an individual’s belief in whether he or she can succeed at a particular activity, plays an integral role in student success,” writes math educator Lynda Colgan of Queen’s University in Ontario, Canada in an article for TVO.org — and that extends to any subject.

“When children are positive about learning and feel able to succeed, they are more likely to be successful.”

Why worry about math now?

“Math is the language of logic,” explains Dr. Jie-Qi (Jackie) Chen , professor of Child Development at the Erikson Institute, a principal investigator of the Early Math Collaborative , and co-author of Big ideas of early mathematics: What teachers of young children need to know .

“Math builds reasoning, which leads to comprehension,” she says. “Developing a mentally organized way of thinking is critical.” To make that happen, Chen says, “We need to provide high-quality math education at an early age.”

She’s right, but are PreK parents and educators listening? Let’s not pretend our children will “catch up” in later grades. In the Program for International Student Assessment, an international assessment that measures 15-year-old students’ reading, mathematics, and science literacy every three years, U.S. test scores in math are embarrassing . In 2012, out of 34 OECD contestants, the USA ranked #27 in math. (We’re #17 in Reading, and #20 in Science.)  In the 2018 PISA math assessment the U.S. ranked #37 out of 76 participating nations. American students are especially weak in “performing mathematics tasks with higher cognitive demand… and interpreting mathematical aspects in real-world problems.” Quite dismally, 26 percent of 15-year-olds in the U.S. fail to reach the PISA baseline Level 2 of mathematics, where they would “begin to demonstrate the skills that will enable them to participate effectively and productively in life.”

So, what can parents do? Without being a total pain-in-the-behind (I’m using a “spatial” term here) you could talk with your child’s preschool director about how they approach math in their daily activities. Talk with them about the value of sorting, measurement, patterns, the language of space, block play, building on innate mental addition skills — and most of all the value of a positive attitude toward math. You might even print out this report: Math Matters: Children’s Mathematical Journeys Start Early   by Deborah J. Stipek and Alan H. Shoenfeld and share it with the preschool staff.

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25 Interesting Math Topics: How to Write a Good Math Essay

writing good math essay

writing good math essay

Mathematics is a fascinating world of numbers, shapes, and patterns. 

Whether you are a student looking to grasp math concepts or someone who finds math intriguing, these topics will spark your curiosity and help you discover the beauty of mathematics straightforwardly and engagingly.

In this article, I will explore interesting math topics that make this subject not only understandable but also enjoyable.

math skills essay

Why Write About Mathematics

First, it helps demystify a subject that many find intimidating. By breaking down complex mathematical concepts into simple, understandable language, we can make math accessible to a wider audience, fostering greater understanding and appreciation.

math book

Second, writing about mathematics allows us to showcase the practical applications of math in everyday life, from managing personal finances to solving real-world problems.

This helps readers recognize the relevance of math and its role in various fields and industries.

Additionally, writing about mathematics can inspire curiosity and a love for learning.

It encourages critical thinking and problem-solving skills, promoting intellectual growth and academic success.

Finally, mathematics is a universal language that transcends cultural and linguistic barriers.

After discussing math topics, we can connect with a global audience, fostering a sense of unity and collaboration in the pursuit of knowledge

 25 Interesting Math Topics to Write On

 Mathematics is a vast and intriguing field, offering a multitude of interesting topics to explore and write about.

Here are 25 such topics that promise to engage both math enthusiasts and those seeking a deeper understanding of this fascinating subject.

1. Fibonacci Sequence: Delve into the mesmerizing world of numbers with this sequence, where each number is the sum of the two preceding ones.

2. Golden Ratio: Explore the ubiquity of the golden ratio in art, architecture, and nature.

3. Prime Numbers: Investigate the mysterious properties of prime numbers and their role in cryptography.

4. Chaos Theory: Understand the unpredictability of chaotic systems and how small changes can lead to drastically different outcomes.

5. Game Theory: Examine the strategies and decision-making processes behind games and real-world situations.

6. Cryptography: Uncover the mathematical principles behind secure communication and encryption.

7. Fractals: Discover the self-replicating geometric patterns that occur in nature and mathematics.

8. Probability Theory: Dive into the world of uncertainty and randomness, where math helps us make informed predictions.

probability theory

9. Number Theory: Explore the properties and relationships of integers, including divisibility and congruence.

10. Geometry of Art: Analyze how geometry and math principles influence art and design.

11. Topology: Study the properties of space that remain unchanged under continuous transformations, leading to the concept of “rubber-sheet geometry.”

12. Knot Theory: Investigate the mathematical study of knots and their applications in various fields.

13. Number Systems: Learn about different number bases, such as binary and hexadecimal, and their significance in computer science.

14. Graph Theory: Explore networks, relationships, and the mathematics of connections.

15. The Monty Hall Problem: Delight in this famous probability puzzle based on a game show scenario.

16. Calculus: Examine the principles of differentiation and integration that underlie a wide range of scientific and engineering applications.

17. The Riemann Hypothesis: Consider one of the most famous unsolved problems in mathematics involving the distribution of prime numbers.

18. Euler’s Identity: Marvel at the beauty of Euler’s equation, often described as the most elegant mathematical formula.

19. The Four-Color Theorem: Uncover the fascinating problem of coloring maps with only four colors without adjacent regions sharing the same color.

20. P vs. NP Problem: Delve into one of the most critical unsolved problems in computer science, addressing the efficiency of algorithms.

21. The Bridges of Konigsberg: Explore a classic problem in graph theory that inspired the development of topology.

22. The Birthday Paradox: Understand the surprising likelihood of shared birthdays in a group.

23. Non-Euclidean Geometry: Step into the world of geometries where Euclid’s parallel postulate doesn’t hold, leading to intriguing alternatives like hyperbolic and elliptic geometry.

24. Perfect Numbers: Learn about the properties of numbers that are the sum of their proper divisors.

25. Zero: The History of Nothing: Trace the historical and mathematical significance of the number zero and its role in the development of mathematics.

How to Write a Good Math Essay

Mathematics essays , though often perceived as daunting, can be a rewarding way to delve into the world of mathematical concepts, problem-solving, and critical thinking.

Whether you are a student assigned to write a math essay or someone who wants to explore math topics in-depth, this guide will provide you with the key steps to write a good math essay that is clear, concise, and engaging.

1. Understanding the Essay Prompt

essay prompts

Before you begin writing, it’s crucial to understand the essay prompt or question.

Analyze the specific topic, the scope of the essay, and any guidelines or requirements provided by your instructor.

Mostly, this initial step sets the direction for your essay and ensures you stay on topic.

2. Research and Gather Information

You need to gather relevant information and resources to write a strong math essay. This includes textbooks, academic papers, and reputable websites.

Make sure to cite your sources properly using a recognized citation style such as APA, MLA, or Chicago.

3. Structuring Your Math Essay

Start with a clear introduction that provides an overview of the topic and the main thesis or argument of your essay. This section should capture the reader’s attention and present a roadmap for what to expect.

The body of your essay is where you present your arguments, explanations, and evidence. Use clear subheadings to organize your ideas. Ensure that your arguments are logical and well-structured.

Begin by defining any important mathematical concepts or terms necessary to understand your topic.

Clearly state your main arguments or theorems. Please support them with evidence, equations, diagrams, or examples.

Explain the logical steps or mathematical reasoning behind your arguments. This can include proofs, derivations, or calculations.

Ensure your writing is clear and free from jargon that might confuse the reader. Explain complex ideas in a way that’s accessible to a broader audience.

Whenever applicable, include diagrams, graphs, or visual aids to illustrate your points. Visual representations can enhance the clarity of your essay.

Summarize your main arguments, restate your thesis, and offer a concise conclusion. Address the significance of your findings and the implications of your research or discussion.

4. Proofreading and Editing

proofreading an essay

Once you’ve written your math essay, take the time to proofread and edit it. Pay attention to grammar, spelling, punctuation, and the overall flow of your writing.

Ensure that your essay is well-organized and free from errors.

Consider seeking feedback from peers or an instructor to gain a fresh perspective.

5. Presentation and Formatting

A well-presented essay is more likely to engage the reader. Follow these formatting guidelines:

  • Use a legible font (e.g., Times New Roman or Arial) in a standard size (12-point).
  • Double-space your essay and include page numbers if required.
  • Create a title page with your name, essay title, course information, and date.
  • Use section headings and subheadings for clarity.
  • Include a reference page to cite your sources appropriately.

6. Mathematical Notation and Symbols

Mathematics relies heavily on notation and symbols. Ensure that you use mathematical notation correctly and consistently.

If you introduce new symbols or terminology, define them clearly for the reader’s understanding.

7. Seek Clarification

If you encounter difficulties or ambiguities in your math essay, don’t hesitate to seek clarification from your instructor or peers.

Discussing complex mathematical concepts with others can help you refine your understanding and improve your essay.

8. Practice and Feedback

Writing math essays, like any skill, improves with practice. The more you write and receive feedback, the better you’ll become.

Take your time with initial challenges. Instead, view them as opportunities for growth and learning.

With dedication and attention to detail, you can craft a math essay that not only conveys your mathematical knowledge but also engages and informs your readers.

Josh Jasen working

Josh Jasen or JJ as we fondly call him, is a senior academic editor at Grade Bees in charge of the writing department. When not managing complex essays and academic writing tasks, Josh is busy advising students on how to pass assignments. In his spare time, he loves playing football or walking with his dog around the park.

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Whether you are contemplating a career in applied math solving real world problems or pure mathematics expanding the realm of what is known (and unknown), make sure your CV or resume includes these 10 skills and abilities.

According to the U.S. Department of Labor , mathematicians should have at least these 10 skills and abilities if they want to succeed in mathematical research, education, or as an industrial mathematician at a Fortune 100 company.

While there are many skills and abilities that make a successful mathematician, employers will review your application materials for a high degree of competency in these skills that show your knowledge as a mathematician. Displaying the ability to exercise information ordering, inductive reasoning, and mathematical reasoning will support your case for employment no matter where you are in your career.

And don’t forget, once you land the interview, share examples of how you used these analytical skills and abilities to succeed.

Mathematics — Using mathematics to solve problems.  

Complex Problem Solving — Identifying complex problems and reviewing related information to develop and evaluate options and implement solutions.  

Critical Thinking — Using logic and reasoning to identify the strengths and weaknesses of alternative solutions, conclusions or approaches to problems.  

Reading Comprehension — Understanding written sentences and paragraphs in work related documents.  

Active Learning — Understanding the implications of new information for both current and future problem-solving and decision-making.  

Mathematical Reasoning — The ability to choose the right mathematical methods or formulas to solve a problem.  

Number Facility — The ability to add, subtract, multiply, or divide quickly and correctly.

Deductive Reasoning — The ability to apply general rules to specific problems to produce answers that make sense.  

Inductive Reasoning — The ability to combine pieces of information to form general rules or conclusions (includes finding a relationship among seemingly unrelated events).

Information Ordering — The ability to arrange things or actions in a certain order or pattern according to a specific rule or set of rules (e.g., patterns of numbers, letters, words, pictures, mathematical operations).

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Essay on Importance of Mathematics in our Daily Life in 100, 200, and 350 words.

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Essay on Importance of Mathematics in our Daily Life

Mathematics is one of the core aspects of education. Without mathematics, several subjects would cease to exist. It’s applied in the science fields of physics, chemistry, and even biology as well. In commerce accountancy, business statistics and analytics all revolve around mathematics. But what we fail to see is that not only in the field of education but our lives also revolve around it. There is a major role that mathematics plays in our lives. Regardless of where we are, or what we are doing, mathematics is forever persistent. Let’s see how maths is there in our lives via our blog essay on importance of mathematics in our daily life. 

This Blog Includes:

Essay on importance of mathematics in our daily life in 100 words , essay on importance of mathematics in our daily life in 200 words, essay on importance of mathematics in our daily life in 350 words.

Mathematics is a powerful aspect even in our day-to-day life. If you are a cook, the measurements of spices have mathematics in them. If you are a doctor, the composition of medicines that make you provide prescription is made by mathematics. Even if you are going out for just some groceries, the scale that is used for weighing them has maths, and the quantity like ‘dozen apples’ has maths in it. No matter the task, one way or another it revolves around mathematics. Everywhere we go, whatever we do, has maths in it. We just don’t realize that. Maybe from now on, we will, as mathematics is an important aspect of our daily life.

Also Read:- Importance of Internet

Mathematics, as a subject, is one of the most important subjects in our lives. Irrespective of the field, mathematics is essential in it. Be it physics, chemistry, accounts, etc. mathematics is there. The use of mathematics proceeds in our daily life to a major extent. It will be correct to say that it has become a vital part of us. Imagining our lives without it would be like a boat without a sail. It will be a shock to know that we constantly use mathematics even without realising the same. 

From making instalments to dialling basic phone numbers it all revolves around mathematics. 

Let’s take an example from our daily life. In the scenario of going out shopping, we take an estimate of hours. Even while buying just simple groceries, we take into account the weight of vegetables for scaling, weighing them on the scale and then counting the cash to give to the cashier. We don’t even realise it and we are already counting numbers and doing calculations. 

Without mathematics and numbers, none of this would be possible.

Hence we can say that mathematics helps us make better choices, more calculated ones throughout our day and hence make our lives simpler. 

Also Read:-   My Aim in Life

Mathematics is what we call a backbone, a backbone of science. Without it, human life would be extremely difficult to imagine. We cannot live even a single day without making use of mathematics in our daily lives. Without mathematics, human progress would come to a halt. 

Maths helps us with our finances. It helps us calculate our daily, monthly as well as yearly expenses. It teaches us how to divide and prioritise our expenses. Its knowledge is essential for investing money too. We can only invest money in property, bank schemes, the stock market, mutual funds, etc. only when we calculate the figures. Let’s take an example from the basic routine of a day. Let’s assume we have to make tea for ourselves. Without mathematics, we wouldn’t be able to calculate how many teaspoons of sugar we need, how many cups of milk and water we have to put in, etc. and if these mentioned calculations aren’t made, how would one be able to prepare tea? 

In such a way, mathematics is used to decide the portions of food, ingredients, etc. Mathematics teaches us logical reasoning and helps us develop problem-solving skills. It also improves our analytical thinking and reasoning ability. To stay in shape, mathematics helps by calculating the number of calories and keeping the account of the same. It helps us in deciding the portion of our meals. It will be impossible to think of sports without mathematics. For instance, in cricket, run economy, run rate, strike rate, overs bowled, overs left, number of wickets, bowling average, etc. are calculated. It also helps in predicting the result of the match. When we are on the road and driving, mathetics help us keep account of our speeds, the distance we have travelled, the amount of fuel left, when should we refuel our vehicles, etc. 

We can go on and on about how mathematics is involved in our daily lives. In conclusion, we can say that the universe revolves around mathematics. It encompasses everything and without it, we cannot imagine our lives. 

Also Read:- Essay on Pollution

Ans: Mathematics is a powerful aspect even in our day-to-day life. If you are a cook, the measurements of spices have mathematics in them. If you are a doctor, the composition of medicines that make you provide prescription is made by mathematics. Even if you are going out for just some groceries, the scale that is used for weighing them has maths, and the quantity like ‘dozen apples’ has maths in it. No matter the task, one way or another it revolves around mathematics. Everywhere we go, whatever we do, has maths in it. We just don’t realize that. Maybe from now on, we will, as mathematics is an important aspect of our daily life.

Ans: Mathematics, as a subject, is one of the most important subjects in our lives. Irrespective of the field, mathematics is essential in it. Be it physics, chemistry, accounts, etc. mathematics is there. The use of mathematics proceeds in our daily life to a major extent. It will be correct to say that it has become a vital part of us. Imagining our lives without it would be like a boat without a sail. It will be a shock to know that we constantly use mathematics even without realising the same. 

From making instalments to dialling basic phone numbers it all revolves around mathematics. Let’s take an example from our daily life. In the scenario of going out shopping, we take an estimate of hours. Even while buying just simple groceries, we take into account the weight of vegetables for scaling, weighing them on the scale and then counting the cash to give to the cashier. We don’t even realise it and we are already counting numbers and doing calculations.

Without mathematics and numbers, none of this would be possible. Hence we can say that mathematics helps us make better choices, more calculated ones throughout our day and hence make our lives simpler.  

Ans: Archimedes is considered the father of mathematics.

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Why is Math Important: Benefits of Learning Math at School

Created: January 11, 2024

Last updated: January 11, 2024

Why is math important is a question worth exploring. Mathematics, a subject beyond mere numbers and formulas, constitutes the core of our existence. Its influence extends far beyond the confines of textbooks, penetrating the very essence of modern life. The topic — why is math the most important subject — also carries weight within the realm of education, which is why kids may be asked to write a why is math important essay in class. 

As we embrace math in education, we enable ourselves to unravel the mysteries of our reality. Through this article, you will discover answers to the question, why is math so important, and understand the many benefits of immersing ourselves in mathematics.

Why Is Math Important for Kids to Learn?

Math plays a significant role in everyone’s educational journey, bringing many benefits beyond just numbers. From the basics like counting and recognizing shapes to more complicated aspects like algebra, geometry, and calculus, studying math grounds students intellectually. At its heart, math teaches discipline and accuracy. 

As people study math, they learn to take logical steps, follow the rules, and pay attention to the details. These skills make their studies easier and help them in other areas of life, teaching them how to approach problems systematically. Math also hones critical thinking and analysis. 

It’s essential to know the answer to the question — why is math important for kids. When faced with math problems, we learn to spot patterns, make connections, and develop hypotheses. This natural problem-solving pathway helps us understand how things work and resolve complex issues. Besides, math literacy is a must-have in a world full of data and tech. Knowing the ins and outs of math gives kids the ability to interpret numbers and make well-thought-out decisions in terms of finance, health, and science. 

Also, math takes students and even teachers to the apex of creativity! When both parties explore numbers, shapes, and equations, they use their imaginations and develop new ways to solve problems and develop ideas. Finally, math encourages collaboration. Group activities and conversations about math help them communicate better, learn together, and make friends. 

Having understood the overview of math’s relevance in people’s lives, let’s delve deeper into why is math important in everyday life for kids.

Math Hones Complex Problem-Solving Skills

Knowing the answer to the ‘why is math important in life’ question enables kids to break down complex problems into smaller components, identify pertinent variables, and use appropriate formulas or methods to arrive at practical solutions. Math equips kids with a structured approach to problem-solving, empowering them to overcome obstacles and adapt to a dynamic world.

The capacity to methodically resolve issues enables them to approach various challenges with unwavering confidence and creativity, whether resolving complex technical troubleshooting issues, streamlining workflows, or interpersonal conflicts. 

It Promotes Critical Thinking

Knowing 5 reasons why math is important reveals math’s role in fostering critical thinking. The journey of solving mathematical problems is crucible for developing critical thinking. As kids immerse themselves in scrutinizing data, solving maze-like word problems, and developing logical strategies, they develop a robust skill — evaluating information from diverse perspectives. 

This ability to see recurring patterns and coherent conclusions is essential to making informed decisions. In debate and dialogue, kids with sharp critical thinking demonstrate the ability to obtain reliable sources, deconstruct complex arguments, and participate meaningfully in discussions. When faced with unexpected circumstances or a whirlwind of rapidly changing scenarios, this honed analytical skill allows them to objectively weigh new information, seamlessly adjust strategies, and deftly navigate the tides of change.

Math Improves Kids Financial Knowledge

Why math is the most important subject is validated within academics, but we can look beyond that. Math knowledge is a vital component of financial literacy, as it provides kids with the understanding and tools to make informed decisions that shape their financial well-being. Mathematics is central in helping individuals develop the essential skills to decipher complex financial concepts. 

From understanding the dynamics of interest rates and the complex effects of investing to evaluating risk and return profiles, mathematics provides the basis for building a solid financial foundation. Using these mathematical insights, kids can create an adequate budget that meets their goals and desires. But financial literacy goes beyond self-interest; it enables them to contribute positively to their communities. 

By making intelligent philanthropic decisions or supporting local businesses, financially savvy kids become agents of change that drive economic growth and community development. Financial literacy provides clarity about a student’s perspective on broader financial issues. 

Using mathematical reasoning, they can engage in informed discussions about public policy, evaluate economic proposals, and make informed choices with far-reaching societies. The combination of mathematics and financial literacy, allowing them to secure their financial future and actively participate in creating a more financially stable and fair society, makes us more confident in answering the question, why is math important?

It Helps Kids Develop Technical Skills

In a digital age where technology permeates every aspect of modern life, the question of ‘why is discrete math important’ is quickly answered. Look at cybersecurity, for example. In 2023, it is among the most sought-after technical skills as companies try to protect their networks and data from breaches and uphold customers’ privacy.

A good understanding of mathematics opens up the ability to understand, analyze and innovate in a complex digital environment. Knowledge of mathematics allows kids to contribute to the development of technology actively. 

As technology evolves and shapes the future, mathematicians are uniquely positioned to drive progress. Using mathematical principles, they confidently explore the digital world, contributing to developing new solutions, advanced applications, and transformative breakthroughs that move society into uncharted territories of technological innovation.

Math Opens The Door to More Career Opportunities

Kids know why math is important and impacts job opportunities because of how many more career paths it offers them. Beyond the bounds of traditional math-oriented roles like engineering and finance, the need for math skills has permeated many industries. Meanwhile, dynamic marketing has used statistical analysis to discover consumer behavior, improve customer segmentation, and drive strategic campaigns. 

As artificial intelligence and automation redefine industries, kids with a solid foundation in mathematics have the adaptability and innovation to thrive in new areas of employment in the future. From harnessing the power of big data to building data-driven narratives, these math-savvy professionals are at the forefront of shaping the future of work.

Learning Math Improves Analytical Skills

Mathematical analysis is crucial for developing analytical thinking, an invaluable skill in our complex, information-saturated world. So why is it important to learn math to improve analytical skills? In an age where navigating massive data sets and deciphering multifaceted challenges is the norm, the ability to discern complex situations and evaluate evidence becomes valuable. 

In a world where career paths and problem-solving paradigms are evolving at an unprecedented rate, the enrichment provided by mathematical and analytical ability is a cornerstone of success. Whether driving an industry into the future or developing innovative solutions to global problems, kids with these skills are built to make a lasting and transformative impact.

Progressive Scientific Discovery

The question — why is math and science important — is a run-off of the belief that math is often the language of science. Math is an indispensable tool for pushing the boundaries of scientific research and inquiry. It is the hidden force behind the breakthrough discoveries that allow scientists to bridge the gap between theoretical concepts and empirical observations.

Clinical trial design and medical analysis are governed by mathematical principles, which aid researchers in evaluating the efficacy of interventions and treatments. Statistical methods rooted in mathematics can provide insight into the effects of new drugs, the spread of diseases, and the impact of public health initiatives. This quantitative approach that improves medical knowledge and saves lives by guiding evidence-based medical practice shows why learning math is important.

It Helps Kids Develop Mental Stamina and Endurance

Facing complex math problems develops mathematical and endurance skills. It promotes mental strength and a will to overcome difficulties. When kids become aware of this, before you point out 5 reasons why math is not important they can already give you countless reasons why math is important. That is because they have gone through the rigors of solving math and now understand that mastery requires dedication, persistence, and a willingness to face failure. 

They develop an inherent resilience beyond mathematics as they solve complex problems and grapple with confusing concepts. This little thing becomes the foundation of personal and professional success. Those who successfully navigate the difficulties of mathematics are better prepared to face the complexity of the modern world. 

As we ponder why math is important in life, we should know that math provides a compass for navigating complexity in a world of information and rapid developments. This article could have still gone ahead to give an extra 10 reasons why math is important as its importance is countless. But you get the point already!

Mastering mathematics nurtures critical thinking, problem-solving abilities, and analytical reasoning — qualities necessary in a world filled with complex challenges and diverse opportunities. These all make us understand why math is important in our daily lives.

But why is learning math important at Brighterly? Brighterly recognizes the transformative power of mathematics and its role in shaping resilient individuals. They provided a platform that supports math understanding and learning. So register now to embark on a journey of discovery, where interactive lessons, engaging activities, and a supportive community await.

Jessica is a a seasoned math tutor with over a decade of experience in the field. With a BSc and Master’s degree in Mathematics, she enjoys nurturing math geniuses, regardless of their age, grade, and skills. Apart from tutoring, Jessica blogs at Brighterly. She also has experience in child psychology, homeschooling and curriculum consultation for schools and EdTech websites.

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Struggling with Math by Gabriela

Gabrielaof Moreno Valley's entry into Varsity Tutor's April 2019 scholarship contest

Struggling with Math by Gabriela - April 2019 Scholarship Essay

Throughout my school years, I’ve always struggled with math. I tended to get really frustrated with it and wanted to give up most of the time. I would feel slow and behind from others which made me dislike the subject. It was hard for me to ask questions because I had that mindset to just try to figure things out on my own which did no good. I would chase the grade not the knowledge but it wasn’t until freshmen year that changed the way I felt about math and about my learning skills. Midway into the school year. I was chosen out of 500 to participate in a math program that would provide tutoring and lessons by students from UCR. I attended a session every Saturday for a few months which helped a group of students strengthen our math skills. I would also present how I would solve for answers, which definitely helped me become more comfortable with public speaking and confident with my problem-solving. This program not only helped me with math but it also allowed me to interact and collaborate with new people, which helped me taught me how to work with other people’s strengths and weaknesses. It took me out of my comfort zone and taught me how to think outside the box. These are definitely some useful skills this opportunity has helped me gain and make stronger. I feel very fortunate to be given this opportunity because it helped me gain new skills and become more of an optimistic person. I became more and more determined throughout the rest of my school years. This opportunity contributed to helping me to deepen my understanding of certain mathematical topics, develop new strategies, and pushed me to become a more open-minded person, as I learned that anything is possible. I have been able to understand the subject better than I have before and now am enrolled in AP Calculus which is less of a struggle for me which I am truly glad that I took advantage of this opportunity.

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Math Extended Essay Topics

Math Extended Essay Topics

Luke MacQuoid

A math extended essay (EE) is a research paper that is written by students who are enrolled in the International Baccalaureate (IB) program as a part of their Diploma. The Extended Essay is self-directed research that aims to provide students with the opportunity to explore a topic of their interest in depth, with a particular focus on the application of math concepts and skills. In order to pass the paper successfully, it is very important to choose the right topic for the Math Extended Essay Topics.

In an Group 5: Mathematics extended essay, students work independently under the guidance of a supervisor to plan and carry out a research project on a math-related topic . This research project should demonstrate the student’s ability to engage in independent research, apply critical thinking and problem-solving skills, and communicate their findings clearly and effectively.

The math extended essay has strict word count in length and should be written in a clear, concise, and formal style, it should also include an introduction, a body of discussion, a conclusion, and a bibliography.

The student is expected to demonstrate the ability to design and conduct a research project, use mathematical concepts and skills to analyze and interpret the data and discuss their findings in the context of relevant mathematical literature.

The math extended essay also allows students to explore a mathematical topic in a way that is not possible in the classroom, it also gives a chance to students to demonstrate their mathematical knowledge and skills and to develop a more global understanding of mathematics.

What’s the Purpose of IB Math Extended Essay?

The purpose of the IB math extended essay is to provide students with the opportunity to engage in independent research and to apply their mathematical knowledge and skills to a topic of their own choosing. The extended essay is designed to give students a chance to explore a topic in depth and to develop the following key skills:

Research skills : Students learn how to design and conduct a research project and how to collect, analyze, and interpret data.

Critical thinking skills : Students learn how to evaluate sources, make a clear and well-supported argument, and solve problems.

Communication skills : Students learn how to write a formal academic paper and how to present their findings and arguments clearly and effectively.

Time management skills:  Students learn how to plan their time effectively and manage their workload.

Subject-specific knowledge : Students get the opportunity to deepen their understanding of a particular area of mathematics and to explore the ways in which mathematical concepts and methods can be applied to solve problems.

The IB Math Extended Essay allows students to research and write about a specific mathematical topic that they find interesting. It also allows students to apply the knowledge and skills they have learned in the classroom to a real-world problem or scenario. 

Completing the extended essay is an important requirement for students to receive the International Baccalaureate Diploma, and it also provides students with the opportunity to develop important research and analytical skills which will be useful in the future, regardless of their chosen career path.

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You can also use our extended essay writers team’s services if you need assistance selecting a topic . Furthermore, we can also help you write your IB extended essay from scratch or edit your draft following the IB criteria.

IB Math Extended Essay Topics

An extended essay in mathematics provides students with an opportunity to explore an area of math that interests them in depth. Here are some potential topics for a math extended essay:

  • Investigating the properties of a specific mathematical function and its applications.
  • Examining the history of a particular mathematical concept or problem
  • Investigating the relationship between different branches of mathematics, such as algebra and geometry.
  • Investigating the properties and application of geometric shapes in different branches of mathematics.
  • Applying mathematical modeling to a real-world problem, such as optimizing logistics or predicting weather patterns.
  • Investigating the use of mathematical methods in computer science, such as complexity theory and algorithms.
  • Exploring the application of number theory in cryptography and information security
  • Investigating the role of statistics in decision-making and forecasting.
  • Investigating the use of mathematical logic in Artificial Intelligence and machine learning
  • Examining the applications of mathematical methods in the field of engineering and physics, such as fluid dynamics and mechanics.
  • Investigating the properties of fractals and their applications in computer graphics and image processing.
  • Exploring the concept of infinity and infinitesimals in calculus and their implications in other branches of mathematics.
  • Investigating the application of graph theory in network analysis and computer science
  • Examining the relationship between topology and other branches of mathematics such as algebra and analysis.
  • Investigating the properties of mathematical knots and their applications in physics and biology.
  • Exploring the application of game theory in economics and decision making
  • Investigating the use of mathematical methods in the field of finance and financial modeling
  • Examining the relationship between chaos theory and other areas of mathematics and science
  • Investigating the use of mathematical methods in the field of cryptography, such as elliptic curve cryptography.
  • Exploring the application of mathematical methods in the study of complexity in natural systems, such as studying the Mandelbrot set in the field of chaos theory.

Remember to check with your teacher or supervisor for guidelines or restrictions and pick a topic that you find interesting and engaging. Also, make sure you have a clear and testable research question as part of the planning and writing process.

Students can use these themes as a starting point for their research and then narrow down their focus to a specific problem or question related to the theme. For example, if a student is interested in investigating the properties of a specific mathematical function, they could begin by researching the history of the function, its key properties, and its applications. 

They could then narrow down their focus to a specific problem or question, such as “How does the shape of the graph of this function change as its parameters are varied?” or “What are the implications of this function’s properties for other areas of mathematics?”

Similarly, if a student is interested in exploring the concept of infinity in calculus, they could begin by researching the history of the concept and its implications in other branches of mathematics. They could then narrow down their focus to a specific question, such as “How does the concept of infinity affect the way we think about limits and continuity in calculus?”

Once a student has a specific problem or question in mind, they can begin to design and conduct their research. This will typically involve collecting data, analyzing it using mathematical methods, and interpreting their findings. They should also read and critically evaluate relevant literature on their topic and use it as a foundation to construct their own arguments.

It’s also important to mention that the research and writing process should be guided by the student’s supervisor and that the student should be able to clearly communicate the research question, methodology, findings, and conclusions in a formal academic paper.

Finally, the extended essay is an opportunity for students to demonstrate their ability to apply mathematical concepts and skills to a real-world problem and to communicate their findings in a clear and effective way, and this can be greatly beneficial for students who wish to pursue further studies or careers in fields related to mathematics and other sciences.

Some Tips to Help You Choose a Relevant Math EE Topic

Choosing a relevant and engaging topic for your math extended essay can be challenging, but by following these tips, you can make the process easier:

Start by identifying your interests: Think about the areas of math that you enjoy and find most interesting. These are likely to be the areas that you will find most engaging to research.

Consider the scope of the topic: Your topic should be narrow enough to be manageable within the 4,000-word limit of the extended essay but broad enough to allow for a thorough investigation.

Look for connections between different areas of mathematics: Often, the most interesting and relevant topics lie at the intersection of different areas of mathematics.

Talk to your supervisor: Your supervisor is a valuable resource and can provide guidance on choosing a relevant and manageable topic. They can also help you refine your research question and give you feedback on your ideas.

Make sure your topic is original: You want your extended essay to be unique and to make a meaningful contribution to the field. Avoid topics that have been extensively researched or that are too similar to other extended essays.

Think about the resources available: Consider whether the resources available to you (books, articles, data, etc.) are sufficient to allow you to conduct the research you want to do.

Consider the audience: Your Extended Essay is read by a group of examiners appointed by the International Baccalaureate, so consider the level of mathematical sophistication of your topic and the language used to communicate it.

Research to find a feasible topic: Once you have some ideas, research them to see if they are feasible, given the time, resources, and supervisor’s expertise you have.

In conclusion,  writing a math extended essay can be a challenging but rewarding experience. The key to success is to choose a relevant and engaging topic that aligns with your interests and allows you to showcase your understanding and skills in mathematics. By considering the tips provided, such as identifying your interests, talking to your supervisor, and making sure your topic is original, you will be able to choose a topic that is both manageable and meaningful. 

One way to ensure the success of their essay is to seek out professional assistance. Professional writers and editors can provide valuable feedback and guidance throughout the research and writing process, helping students to improve their understanding of the topic and to communicate their findings more effectively.

It could be a good idea to consider looking into reputable academic writing services where you can get help with your math extended essay and improve your chances of getting a good grade. They can also help ensure that the paper meets academic standards and requirement. However, always be sure to evaluate the source carefully and do your due diligence, as not all academic writing services are created equal.

Remember that the extended essay is an opportunity to conduct independent research and apply critical thinking and problem-solving skills to a topic of your own choosing. By following a systematic and well-structured approach, you will be able to successfully complete your math extended essay and demonstrate your knowledge and understanding of the subject.

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Luke MacQuoid has extensive experience teaching English as a foreign language in Japan, having worked with students of all ages for over 12 years. Currently, he is teaching at the tertiary level. Luke holds a BA from the University of Sussex and an MA in TESOL from Lancaster University, both located in England. As well to his work as an IB Examiner and Master Tutor, Luke also enjoys sharing his experiences and insights with others through writing articles for various websites, including extendedessaywriters.com blog

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Math In Everyday Life (Essay Sample)

Most students have the history of struggling with mathematics assignments which make them wonder if they will ever apply the knowledge in real world life. Teachers and parents admit having been asked about the relevance of mathematics in life. Some often respond that mathematics is necessary for most of the jobs and it enhances critical thinking skills of an individual. Such responses may be good but fail to address immediate needs of a student. There are various everyday practical applications of mathematics.

The most common and essential application of mathematics in daily life is in financial management like spending, investing and saving. The modern world is money-driven and therefore, demands knowledge in mathematics to help in various calculations. Inability to add, subtract, multiply or divide will result in serious difficulties when handling money. One requires mathematical skills beyond the basic arithmetic concepts because complex algebra is necessary when calculating interest rates for loans and investments. Even those who manage their own money will often require dealing with loans and investment. Besides, one should be familiar with exponential growth calculations to plan for future projections especially in spending of money.

Mathematics concepts are very significant in the management of time. Time factor affects everyone around the world. Time is, therefore, a very valuable asset that cannot be ignored. The modern competitive world demands very proper planning of time to avoid lagging behind as others make progress. Failing to keep schedule results in accumulation of things not done in the required time frame. Mathematics comes into play when organizing to-do lists where the rating of tasks varies depending on the priority and urgency. Organising a schedule is, therefore, beyond the simple knowledge of reading the clock or calendar; more application of mathematics is necessary.

In addition, mathematics is applied every day in the grocery stores. This sector requires an accurate use of mathematics knowledge such as estimation, multiplications, and percentages. The sellers apply mathematics in the calculation of price per unit, estimating the percentage of discounts offered, measuring the weight of products and estimating of the net price required to sell or buy a product. Mathematics in the grocery stores is the best example of its application in sectors affecting everyone and every day.

Closely related is the application of mathematics in the kitchen. Recipes for different foods vary either in terms of ingredients used or simply by the number of ingredients. Preparing the recipes requires a step-by-step operation steps similar to algorithms. Some of the mathematical applications include: measuring of ingredients, converting of temperatures necessary in the preparation of different foods, calculation of cooking duration for each item, and calculation of ratio and proportions, especially in baking.

Another daily application of mathematics happens in traveling. It helps to take away risks associated with traveling. Before one starts a journey, they have to estimate different factors such as the amount of fuel to be used per distance and hour. This is crucial for long distance travelers since there is a risk of exhausting the available fuel and remain stranded on the road. Mathematics still come into play when paying for tolls, determining the tire pressure, checking the numbers of exit and others. Besides, mathematics is crucial when using maps during the road trip in the absence of GPS and Google Maps. One should find their current location and how to move to the destination. Basic mathematics fundamentals are applied in such instances.

Basing on the above daily applications and others, it is evident that knowledge in mathematics is necessary and relevant to everyone regardless of the occupations and status they occupy in the society.

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Guest Essay

When Your Technical Skills Are Eclipsed, Your Humanity Will Matter More Than Ever

A graphic depicting a door being opened to  reveals a handshake, a cup of a coffee, a briefcase and a swirl of colors.

By Aneesh Raman and Maria Flynn

Mr. Raman is a work force expert at LinkedIn. Ms. Flynn is the president of Jobs for the Future.

There have been just a handful of moments over the centuries when we have experienced a huge shift in the skills our economy values most. We are entering one such moment now. Technical and data skills that have been highly sought after for decades appear to be among the most exposed to advances in artificial intelligence. But other skills, particularly the people skills that we have long undervalued as soft, will very likely remain the most durable. That is a hopeful sign that A.I. could usher in a world of work that is anchored more, not less, around human ability.

A moment like this compels us to think differently about how we are training our workers, especially the heavy premium we have placed on skills like coding and data analysis that continue to reshape the fields of higher education and worker training. The early signals of what A.I. can do should compel us to think differently about ourselves as a species. Our abilities to effectively communicate, develop empathy and think critically have allowed humans to collaborate, innovate and adapt for millenniums. Those skills are ones we all possess and can improve, yet they have never been properly valued in our economy or prioritized in our education and training. That needs to change.

In today’s knowledge economy, many students are focused on gaining technical skills because those skills are seen as the most competitive when it comes to getting a good job. And for good reason. For decades, we have viewed those jobs as future-proof, given the growth of technology companies and the fact that engineering majors land the highest-paying jobs .

The number of students seeking four-year degrees in computer science and information technology shot up 41 percent between the spring of 2018 and the spring of 2023, while the number of humanities majors plummeted. Workers who didn’t go to college and those who needed additional skills and wanted to take advantage of a lucrative job boom flocked to dozens of coding boot camps and online technical programs.

Now comes the realization of the power of generative A.I., with its vast capabilities in skills like writing, programming and translation. (Microsoft, which owns LinkedIn, is a major investor in the technology.) LinkedIn researchers recently looked at which skills any given job requires and then identified over 500 likely to be affected by generative A.I. technologies. They then estimated that 96 percent of a software engineer’s current skills — mainly proficiency in programming languages — can eventually be replicated by A.I. Skills associated with jobs like legal associates and finance officers will also be highly exposed.

In fact, given the broad impact A.I. is set to have, it is quite likely to affect all of our work to some degree or another.

We believe there will be engineers in the future, but they will most likely spend less time coding and more time on tasks like collaboration and communication. We also believe that there will be new categories of jobs that emerge as a result of A.I.’s capabilities — just like we’ve seen in past moments of technological advancement — and that those jobs will probably be anchored increasingly around people skills.

Circling around this research is the big question emerging across so many conversations about A.I. and work, namely: What are our core capabilities as humans?

If we answer that question from a place of fear about what’s left for people in the age of A.I., we can end up conceding a diminished view of human capability. Instead, it’s critical for us all to start from a place that imagines what’s possible for humans in the age of A.I. When you do that, you find yourself focusing quickly on people skills that allow us to collaborate and innovate in ways technology can amplify but never replace. And you find yourself — whatever the role or career stage you’re in — with agency to better manage this moment of historic change.

Communication is already the most in-demand skill across jobs on LinkedIn today. Even experts in A.I. are observing that the skills we need to work well with A.I. systems, such as prompting, are similar to the skills we need to communicate and reason effectively with other people.

Over 70 percent of executives surveyed by LinkedIn last year said soft skills were more important to their organizations than highly technical A.I. skills. And a recent Jobs for the Future survey found that 78 percent of the 10 top-employing occupations classified uniquely human skills and tasks as “important” or “very important.” These are skills like building interpersonal relationships, negotiating between parties and guiding and motivating teams.

Now is the time for leaders, across sectors, to develop new ways for students to learn that are more directly, and more dynamically, tied to where our economy is going, not where it has been. Critically, that involves bringing the same level of rigor to training around people skills that we have brought to technical skills.

Colleges and universities have a critical role to play. Over the past few decades, we have seen a prioritization of science and engineering, often at the expense of the humanities. That calibration will need to be reconsidered.

Those not pursuing a four-year degree should look for those training providers that have long emphasized people skills and are invested in social capital development.

Employers will need to be educators not just around A.I. tools but also on people skills and people-to-people collaboration. Major employers like Walmart and American Airlines are already exploring ways to put A.I. in the hands of employees so they can spend less time on routine tasks and more time on personal engagement with customers.

Ultimately, for our society, this comes down to whether we believe in the potential of humans with as much conviction as we believe in the potential of A.I. If we do, it is entirely possible to build a world of work that not only is more human but also is a place where all people are valued for the unique skills they have, enabling us to deliver new levels of human achievement across so many areas that affect all of our lives, from health care to transportation to education. Along the way, we could meaningfully increase equity in our economy, in part by addressing the persistent gender gap that exists when we undervalue skills that women bring to work at a higher percentage than men.

Almost anticipating this moment a few years ago, Minouche Shafik, who is now the president of Columbia University, said: “In the past, jobs were about muscles. Now they’re about brains, but in the future, they’ll be about the heart.”

The knowledge economy that we have lived in for decades emerged out of a goods economy that we lived in for millenniums, fueled by agriculture and manufacturing. Today the knowledge economy is giving way to a relationship economy, in which people skills and social abilities are going to become even more core to success than ever before. That possibility is not just cause for new thinking when it comes to work force training. It is also cause for greater imagination when it comes to what is possible for us as humans not simply as individuals and organizations but as a species.

Aneesh Raman is a vice president and work force expert at LinkedIn. Maria Flynn is the president of Jobs for the Future.

The Times is committed to publishing a diversity of letters to the editor. We’d like to hear what you think about this or any of our articles. Here are some tips . And here’s our email: [email protected] .

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An earlier version of this article misstated the group surveyed in a poll on worker skills. The respondents were executives in the United States, not executives at LinkedIn.

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Science News

Social media harms teens’ mental health, mounting evidence shows. what now.

Understanding what is going on in teens’ minds is necessary for targeted policy suggestions

A teen scrolls through social media alone on her phone.

Most teens use social media, often for hours on end. Some social scientists are confident that such use is harming their mental health. Now they want to pinpoint what explains the link.

Carol Yepes/Getty Images

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By Sujata Gupta

February 20, 2024 at 7:30 am

In January, Mark Zuckerberg, CEO of Facebook’s parent company Meta, appeared at a congressional hearing to answer questions about how social media potentially harms children. Zuckerberg opened by saying: “The existing body of scientific work has not shown a causal link between using social media and young people having worse mental health.”

But many social scientists would disagree with that statement. In recent years, studies have started to show a causal link between teen social media use and reduced well-being or mood disorders, chiefly depression and anxiety.

Ironically, one of the most cited studies into this link focused on Facebook.

Researchers delved into whether the platform’s introduction across college campuses in the mid 2000s increased symptoms associated with depression and anxiety. The answer was a clear yes , says MIT economist Alexey Makarin, a coauthor of the study, which appeared in the November 2022 American Economic Review . “There is still a lot to be explored,” Makarin says, but “[to say] there is no causal evidence that social media causes mental health issues, to that I definitely object.”

The concern, and the studies, come from statistics showing that social media use in teens ages 13 to 17 is now almost ubiquitous. Two-thirds of teens report using TikTok, and some 60 percent of teens report using Instagram or Snapchat, a 2022 survey found. (Only 30 percent said they used Facebook.) Another survey showed that girls, on average, allot roughly 3.4 hours per day to TikTok, Instagram and Facebook, compared with roughly 2.1 hours among boys. At the same time, more teens are showing signs of depression than ever, especially girls ( SN: 6/30/23 ).

As more studies show a strong link between these phenomena, some researchers are starting to shift their attention to possible mechanisms. Why does social media use seem to trigger mental health problems? Why are those effects unevenly distributed among different groups, such as girls or young adults? And can the positives of social media be teased out from the negatives to provide more targeted guidance to teens, their caregivers and policymakers?

“You can’t design good public policy if you don’t know why things are happening,” says Scott Cunningham, an economist at Baylor University in Waco, Texas.

Increasing rigor

Concerns over the effects of social media use in children have been circulating for years, resulting in a massive body of scientific literature. But those mostly correlational studies could not show if teen social media use was harming mental health or if teens with mental health problems were using more social media.

Moreover, the findings from such studies were often inconclusive, or the effects on mental health so small as to be inconsequential. In one study that received considerable media attention, psychologists Amy Orben and Andrew Przybylski combined data from three surveys to see if they could find a link between technology use, including social media, and reduced well-being. The duo gauged the well-being of over 355,000 teenagers by focusing on questions around depression, suicidal thinking and self-esteem.

Digital technology use was associated with a slight decrease in adolescent well-being , Orben, now of the University of Cambridge, and Przybylski, of the University of Oxford, reported in 2019 in Nature Human Behaviour . But the duo downplayed that finding, noting that researchers have observed similar drops in adolescent well-being associated with drinking milk, going to the movies or eating potatoes.

Holes have begun to appear in that narrative thanks to newer, more rigorous studies.

In one longitudinal study, researchers — including Orben and Przybylski — used survey data on social media use and well-being from over 17,400 teens and young adults to look at how individuals’ responses to a question gauging life satisfaction changed between 2011 and 2018. And they dug into how the responses varied by gender, age and time spent on social media.

Social media use was associated with a drop in well-being among teens during certain developmental periods, chiefly puberty and young adulthood, the team reported in 2022 in Nature Communications . That translated to lower well-being scores around ages 11 to 13 for girls and ages 14 to 15 for boys. Both groups also reported a drop in well-being around age 19. Moreover, among the older teens, the team found evidence for the Goldilocks Hypothesis: the idea that both too much and too little time spent on social media can harm mental health.

“There’s hardly any effect if you look over everybody. But if you look at specific age groups, at particularly what [Orben] calls ‘windows of sensitivity’ … you see these clear effects,” says L.J. Shrum, a consumer psychologist at HEC Paris who was not involved with this research. His review of studies related to teen social media use and mental health is forthcoming in the Journal of the Association for Consumer Research.

Cause and effect

That longitudinal study hints at causation, researchers say. But one of the clearest ways to pin down cause and effect is through natural or quasi-experiments. For these in-the-wild experiments, researchers must identify situations where the rollout of a societal “treatment” is staggered across space and time. They can then compare outcomes among members of the group who received the treatment to those still in the queue — the control group.

That was the approach Makarin and his team used in their study of Facebook. The researchers homed in on the staggered rollout of Facebook across 775 college campuses from 2004 to 2006. They combined that rollout data with student responses to the National College Health Assessment, a widely used survey of college students’ mental and physical health.

The team then sought to understand if those survey questions captured diagnosable mental health problems. Specifically, they had roughly 500 undergraduate students respond to questions both in the National College Health Assessment and in validated screening tools for depression and anxiety. They found that mental health scores on the assessment predicted scores on the screenings. That suggested that a drop in well-being on the college survey was a good proxy for a corresponding increase in diagnosable mental health disorders. 

Compared with campuses that had not yet gained access to Facebook, college campuses with Facebook experienced a 2 percentage point increase in the number of students who met the diagnostic criteria for anxiety or depression, the team found.

When it comes to showing a causal link between social media use in teens and worse mental health, “that study really is the crown jewel right now,” says Cunningham, who was not involved in that research.

A need for nuance

The social media landscape today is vastly different than the landscape of 20 years ago. Facebook is now optimized for maximum addiction, Shrum says, and other newer platforms, such as Snapchat, Instagram and TikTok, have since copied and built on those features. Paired with the ubiquity of social media in general, the negative effects on mental health may well be larger now.

Moreover, social media research tends to focus on young adults — an easier cohort to study than minors. That needs to change, Cunningham says. “Most of us are worried about our high school kids and younger.” 

And so, researchers must pivot accordingly. Crucially, simple comparisons of social media users and nonusers no longer make sense. As Orben and Przybylski’s 2022 work suggested, a teen not on social media might well feel worse than one who briefly logs on. 

Researchers must also dig into why, and under what circumstances, social media use can harm mental health, Cunningham says. Explanations for this link abound. For instance, social media is thought to crowd out other activities or increase people’s likelihood of comparing themselves unfavorably with others. But big data studies, with their reliance on existing surveys and statistical analyses, cannot address those deeper questions. “These kinds of papers, there’s nothing you can really ask … to find these plausible mechanisms,” Cunningham says.

One ongoing effort to understand social media use from this more nuanced vantage point is the SMART Schools project out of the University of Birmingham in England. Pedagogical expert Victoria Goodyear and her team are comparing mental and physical health outcomes among children who attend schools that have restricted cell phone use to those attending schools without such a policy. The researchers described the protocol of that study of 30 schools and over 1,000 students in the July BMJ Open.

Goodyear and colleagues are also combining that natural experiment with qualitative research. They met with 36 five-person focus groups each consisting of all students, all parents or all educators at six of those schools. The team hopes to learn how students use their phones during the day, how usage practices make students feel, and what the various parties think of restrictions on cell phone use during the school day.

Talking to teens and those in their orbit is the best way to get at the mechanisms by which social media influences well-being — for better or worse, Goodyear says. Moving beyond big data to this more personal approach, however, takes considerable time and effort. “Social media has increased in pace and momentum very, very quickly,” she says. “And research takes a long time to catch up with that process.”

Until that catch-up occurs, though, researchers cannot dole out much advice. “What guidance could we provide to young people, parents and schools to help maintain the positives of social media use?” Goodyear asks. “There’s not concrete evidence yet.”

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Title: self-discover: large language models self-compose reasoning structures.

Abstract: We introduce SELF-DISCOVER, a general framework for LLMs to self-discover the task-intrinsic reasoning structures to tackle complex reasoning problems that are challenging for typical prompting methods. Core to the framework is a self-discovery process where LLMs select multiple atomic reasoning modules such as critical thinking and step-by-step thinking, and compose them into an explicit reasoning structure for LLMs to follow during decoding. SELF-DISCOVER substantially improves GPT-4 and PaLM 2's performance on challenging reasoning benchmarks such as BigBench-Hard, grounded agent reasoning, and MATH, by as much as 32% compared to Chain of Thought (CoT). Furthermore, SELF-DISCOVER outperforms inference-intensive methods such as CoT-Self-Consistency by more than 20%, while requiring 10-40x fewer inference compute. Finally, we show that the self-discovered reasoning structures are universally applicable across model families: from PaLM 2-L to GPT-4, and from GPT-4 to Llama2, and share commonalities with human reasoning patterns.

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CISCE ISC Math Exam 2024: Students say paper was tricky and lengthy

Students from various schools share mixed reactions on isc mathematics papers, ranging from application-based to tricky questions..

The ISC (Class 12) students who appeared in the mathematics examination on Tuesday found the question paper to be moderately difficult, tricky and tad lengthy too.

CISCE ISC Math Exam 2024: Students say paper was tricky and lengthy(Mourya/ Hindustan Times)

In Lucknow, students found the multiple-choice questions a bit tricky but section C was comparatively easy. The questions of section A covered the whole syllabus and was based on the specimen paper provided by the council.

Kashvi Pandey, a class 12 student of La Martiniere Girls College said, “The paper was very much application based. Despite general apprehension, we were able to give our best. Our pre board preparation helped us to understand and solve the questions quicker.”

Vaidehi Baranwal from LMGC said, “The paper was completely based on higher order thinking and practical application of the formulae. It was surely lengthy, additionally, I did find a few questions to be tricky but nonetheless the standard of ISC was maintained. The objective questions took longer than expected due to which I couldn't manage my time well but at the end I was able to complet all the questions.”

Chhavi, a student of City Montessori School, station road branch said: “The paper was mainly Application based and it required in-depth knowledge to solve certain graphical MCQs. It showcased a wide range of mathematical concepts at moderate difficulty level. A well-balanced distribution of topics contributed to a fair and thorough assessment.”

Dev Chaturvedi, another student from the same school said, “ The medium difficulty Mathematics paper provided a solid evaluation of fundamental concepts without overwhelming complexity. The statements were too complex to understand.”

Anshika Gupta, another student said, “Paper was moderate to difficult. 1 marks questions were application based and could not be solved until some 4-mark questions were from the Previous Year Question papers.”

Pranam Goyal, a student said: “The paper was of moderate level. The questions were application based. The in-depth knowledge was tested from MCQs and multiple concepts were questioned through various questions. Few questions were however lengthy. Overall the paper needed a thoughtful approach and problem solving skills.”

Piyush Tripathi, a student said: “The paper was very different from previous year papers. The paper was application based which made it a bit difficult. On a conclusion basis in depth knowledge was required for the paper.”

Shaurya Jaiswal said, “The paper was application based and involved usage of in-depth knowledge. There were many case studies in the paper which needed proper analysis. The paper was quite lengthy but doable. On a complete basis the paper was of moderate level.” Srishti Singh said, “The questions this year were very twisted and not direct. Somewhat deceptive language was used. The change in the pattern of the paper and its application-based nature was new to students and children are still adapting to this.”

Tanishka Sharma of St Joseph College said, “The ISC mathematics paper was overall moderate but a bit lengthy. The mcqs very conceptual and to the point but apart from that the rest of the paper was really calculative.”

Aarav Shukla of St Joseph College said, “The paper was more challenging compared to the previous year.It had a total of 22 questions divided into three sections.Out of the last two optional sections, Section C was easier. All questions were from the prescribed syllabus.”

Vasu, a student of City Montessori School, LDA branch said, “There were a few tricky questions." Another science Student Varnit said, “The paper was neither tough nor easy and I enjoyed solving it.” Jay said, “The paper needed better time management." Vineeta Kamran, the principal, praised the efforts made by the teachers that made students comfortable.

By and large the students found paper moderate and a few students are expecting full marks. However, the question on the topic Probability was tricky. The paper was lengthy but the students managed to finish the paper on time, said Tanmay Pandey and Anant Anand of Class XII here at CMS RDSO Campus.

Students of Strawberry Fields High School in Sector 26, Chandigarh overall felt that the mathematics exam had gone as per their expectations. Vithal said that the exam wasn't hard but the way that the questions tested their understanding and application of the concepts made the exam a bit lengthy. Sashit Sapra added that the exam was well balanced, and all topics were covered almost equally from Chandigarh.

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Rajeev Mullick is a Special Correspondent, he writes on education, telecom and heads city bureau at Lucknow. Love travelling ...view detail

math skills essay

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